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MAA SL Math Test: Sequences, Series & Binomial Theorem (Paper 1)
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MAA SL Test on Sequences and the Binomial Theorem by Christos Nikolaidis Date: 27 November 2019 Marks: /40 Paper 1: without GDC Name of student: __________________________________________________ 1. [Maximum mark: 5] Consider the arithmetic sequence -6, -3, 0, 3, 6, ! Find (a) the twentieth term. [2] (b) the sum of the first twenty terms. [2] (c) the product of the first twenty terms. 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Turn over 1 2. [Maximum mark: 6] Consider the geometric sequence a , 8, 16, b , ! (a) Write down the value of a and the value of b . (b) Find the sum of the first ten terms in the form 2 m − 4 , where m is an integer to (c) [2] be determined. [3] Explain why the sum to infinity does not exist. 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[Maximum mark: 6] Expand (a) (2 x − 3 y ) 3 [3] (b) (1 − 3 ) 4 in the form a + b 3 , where a, b ∈ Z . 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Turn over 3 4. [Maximum mark: 6] The fifth term of an arithmetic sequence is 31 while the ninth term is 59. Find the sum of the first two terms. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 4 5. [Maximum mark: 5] (a) Find the sum of the infinite geometric series 0.2, 0.2 2 , 0.2 3 , ... in the form (b) 1 where a is an integer. a [2] Find the sum of the infinite geometric series 3 , − 3 × 0.2, 3 × 0.2 2 , ... in the form a where a , b are integers. b [3] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Turn over 5 6. [Maximum mark: 5] Find 18 (a) ∑ (10k + 5) [3] k =1 18 (b) ∑ (10k + 5) [2] k =3 .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 6 7. [Maximum mark: 7] The terms x, y, x+ y are consecutive in an arithmetic sequence. The terms 4, x , y are consecutive in a geometric sequence. (a) Find the value of x and the value of y [5] (b) Confirm that the results are true. [2] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 7 8 MAA SL Test on Sequences and the Binomial Theorem by Christos Nikolaidis Date: 27 November 2019 Marks: /40 Paper 2: with GDC Name of student: __________________________________________________ 1. [Maximum mark: 5] Find the sum 17+26+35+44+ ‧‧‧ +152 .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Turn over 1 2. [Maximum mark: 9] The tenth term of an arithmetic sequence is 98 while the sum of the first ten terms is 575. Find (a) The first term u1 [2] (b) The common difference d . [2] (c) The sum of the first 18 terms. [2] (d) Find the first term which exceeds 500. 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(please continue on the following page) 2 .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Turn over 3 3. [Maximum mark: 8] Consider the geometric sequence 6, 2 3, 2, ! (a) Confirm that the first 3 terms are in geometric sequence and state the ratio. [2] (b) Find the sum of the infinite series in the form a + b 3 [2] (c) Find the 10th term correct in 3 significant figures. [2] (d) Find the sum of the first 10 terms correct in 4 significant figures. 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[Maximum mark: 5] ( ) 7 (a) Find the coefficient of the x 6 in the expansion of 3 x 2 − 5 (b) Find the coefficient of the x 8 in the expansion of x 2 3 x 2 − 5 ( [3] ) 7 [2] .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. Turn over 5 5. [Maximum mark: 5] 6 k The constant term in the expansion of 3 x 2 + is 84375. x Find the possible values of k . .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. .................................................................................................................................. 6 6. [Maximum mark: 8] Tzannis invests 800€ at 8% per year (compounded yearly). Alexia invests 780€ at 8% per year compounded monthly. Panos invests 500€ at 10% per year (compounded yearly). (a) Find whether Tzannis or Alexia receives more money after 10 years. (b) Tzannis estimates that he will receive more than 3000€ after n complete years. Find the value of n . (c) [4] [2] Panos receives more money that Tzannis after m complete years. Find the value of m . 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