SL_Arithmetic, Geometric, Exponents, and Logs [45 marks]
QUESTION 1
The first three terms of a geometric sequence are lnπ₯16 , lnπ₯ 8 , lnπ₯ 4 , for π₯ > 0.
[3]
Find the common ratio.
QUESTION 2
Solve the equation 2 ln π₯ = ln 9 + 4. Give your answer in the form π₯ = ππ π where π, π ∈ β€+ .
[5]
QUESTION 3
The πth term of an arithmetic sequence is given by π’π = 15 − 3π.
(a) State the value of the first term, π’1 .
[1]
(b) Given that the πth term of this sequence is −33, find the value of π.
[2]
(c) Find the common difference, π.
[2]
QUESTION 4
Consider an arithmetic sequence where π’8 = π8 = 8.
[5]
Find the value of the first term, π’1 , and the value of the common difference, π.
QUESTION 5
The first three terms of an arithmetic sequence are π’1 , 5π’1 − 8 and 3π’1 + 8.
(a) Show that π’1 = 4.
(b) Prove that the sum of the first π terms of this arithmetic sequence is a square number.
[2]
[4]
QUESTION 6
In an arithmetic sequence, π’2 = 5 and π’3 = 11.
(a) Find the common difference.
[2]
(b) Find the first term.
[2]
(c) Find the sum of the first 20 terms.
[2]
QUESTION 7
1
Consider the series ln π₯ + π ln π₯ + 3 ln π₯ + β―, where π₯ ∈ β, π₯ > 1 and π ∈ β, π ≠ 0.
Consider the case where the series is geometric.
(a)
(i)
Show that π = ±
1
.
√3
(ii) Given that π > 0 and π∞ = 3 + √3, find the value of π₯.
[2]
[3]
Now consider the case where the series is arithmetic with common difference π.
(b)
2
(i) Show that π = 3.
[3]
(ii) Write down π in the form π ln π₯, where π ∈ β.
[1]
(iii) The sum of the first π terms of the series is −3 ln π₯.
[6]
Find the value of π.
HL_Arithmetic, Geometric, Exponents, and Logs [21 marks]
QUESTION 1
1
Show that logπ 2 π₯ = 2 logπ π₯ where π, π₯ ∈ β+ .
[2]
QUESTION 2
Find the solution of log2 π₯ − log2 5 = 2 + log2 3.
[4]
QUESTION 3
Solve the equation log2 (π₯ + 3) + log2 (π₯ − 3) = 4.
[5]
QUESTION 4
The 1st, 4th and 8th terms of an arithmetic sequence, with common difference π, π ≠ 0, are the first
three terms of a geometric sequence, with common ratio π. Given that the 1st term of both sequences
is 9 find
(a) the value of π;
[4]
(b) the value of π;
[1]
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