Mathematical Investigations IV S&S Unit test Name ______________ You may use a TI-30 Calculator on this exam. Justify all your work. 1. (2 pts each) Given the numbers 8 and 128, find their: a) arithmetic mean b) geometric mean 8 128 68 2 2. 8 128 32 Consider the series 4 7 5 8 6 9 a. (4 pts) Write the series in 62 k (k 3) k 4 62 65 . notation. 59 or (k 3)(k 6) k 1 b. (5 pts) Use appropriate formulas to evaluate the series. 59 59 k 1 k 1 (k 3)(k 6) (k 2 9k 18) 59 59 59 k 1 k 1 k 1 k 2 9 k 18 59 60 119 59 60 9 59 18 6 2 87, 202 c) harmonic mean 2 8 128 256 15.0588 8 128 17 Mathematical Investigations IV 3. S&S Unit test (3 pts each) If a series has a first term of 16 and the sum of its first three terms is 76, find: a. The common ratio if the series is geometric. 16 16r 16r 2 76 16r 16r 2 60 4r 4r 2 15 4r 2 4r 15 0 (2r 3)(2r 5) 0 3 5 r or 2 2 b. The common difference if the series is arithmetic. 16 (16 d ) (16 2d ) 76 3d 28 28 d 3 4. Name ______________ (3 pts each) Write out the first four terms of: a. The sequence rn n1 if if n 1 5 rn 3 if n 2 r 2r if n 2 n2 n 1 5, 3, 7,1 92 b. The series (2 k 1 2, 4, 4, 0 k 4k ) Mathematical Investigations IV 5. S&S Unit test Name ______________ n (4 pts) Let S n ak . Suppose that Sn n 2 7n 1 for all positive integers n. k 1 Determine the first three terms of the sequence ak k 1 . a1 S1 12 7 1 1 5 a1 a2 S 2 22 7 2 1 9 a2 4 a1 a2 a3 S3 32 7 3 1 11 a3 2 5, 4, 2 6. (3 pts each) Write: a. A recursive formula for the sequence defined by bn 18 . 3n if n 1 6 bn 1 3 bn 1 if n 1 if n 1 7 b. An explicit formula for the sequence defined by: cn cn1 2 if n 1 cn 7 2(n 1) or cn 2n 5 Mathematical Investigations IV S&S Unit test Name ______________ n 7. Let S n represent the nth partial sum of the series 4k k 1 8 2 1 . a. (4 pts) Use mathematical induction to prove that S n 4 1 Base Case: (n = 1): 4k k 1 8 2 8 , 1 3 S1 4 4 4 8 4 , so base case holds. 2 1 1 3 3 n Inductive Step: I Suppose that for some positive integer n, 4k k 1 Then, n 1 4k k 1 8 2 4 for all positive integers n. 2n 1 8 8 8 8 8 2 1 3 15 35 4n 1 4( n 1) 2 1 4 8 4 , by induction hypothesis 2n 1 4( n 1) 2 1 4 8 4 2 2n 1 4(n 2n 1) 1 4 8 4 2 2 n 1 4 n 8n 3 4 8 4 2n 1 (2n 1)(2n 3) 4(2n 3) 8 4 2 (2n 1)(2n 3) 4(n 2n 1) 1 8n 12 8 4 2 (2n 1)(2n 3) 4(n 2n 1) 1 8n 4 4 (2n 1)(2n 3) 4(2n 1) 4 (2n 1)(2n 3) 4 4 (2n 3) 4 4 (2(n 1) 1) So Statement holds for all positive integers n. 8 2 1 4 4 . 2n 1 Mathematical Investigations IV S&S Unit test Name ______________ b. (2 pts) Use the formula in part a. to determine the value of 4k k 1 4k k 1 8 2 1 lim S n n 4 lim 4 n 2n 1 40 4 8 2 1 .