SL_Arithmetic, Geometric, Exponents, and Logs [45 marks]
QUESTION 1
The first three terms of a geometric sequence are ln𝑥16 , ln𝑥 8 , ln𝑥 4 , for 𝑥 > 0.
[3]
Find the common ratio.
QUESTION 2
Solve the equation 2 ln 𝑥 = ln 9 + 4. Give your answer in the form 𝑥 = 𝑝𝑒 𝑞 where 𝑝, 𝑞 ∈ ℤ+ .
[5]
QUESTION 3
The 𝑛th term of an arithmetic sequence is given by 𝑢𝑛 = 15 − 3𝑛.
(a) State the value of the first term, 𝑢1 .
[1]
(b) Given that the 𝑛th term of this sequence is −33, find the value of 𝑛.
[2]
(c) Find the common difference, 𝑑.
[2]
QUESTION 4
Consider an arithmetic sequence where 𝑢8 = 𝑆8 = 8.
[5]
Find the value of the first term, 𝑢1 , and the value of the common difference, 𝑑.
QUESTION 5
The first three terms of an arithmetic sequence are 𝑢1 , 5𝑢1 − 8 and 3𝑢1 + 8.
(a) Show that 𝑢1 = 4.
(b) Prove that the sum of the first 𝑛 terms of this arithmetic sequence is a square number.
[2]
[4]
QUESTION 6
In an arithmetic sequence, 𝑢2 = 5 and 𝑢3 = 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 𝑥 + 𝑝 ln 𝑥 + 3 ln 𝑥 + ⋯, where 𝑥 ∈ ℝ, 𝑥 > 1 and 𝑝 ∈ ℝ, 𝑝 ≠ 0.
Consider the case where the series is geometric.
(a)
(i)
Show that 𝑝 = ±
1
.
√3
(ii) Given that 𝑝 > 0 and 𝑆∞ = 3 + √3, find the value of 𝑥.
[2]
[3]
Now consider the case where the series is arithmetic with common difference 𝑑.
(b)
2
(i) Show that 𝑝 = 3.
[3]
(ii) Write down 𝑑 in the form 𝑘 ln 𝑥, where 𝑘 ∈ ℚ.
[1]
(iii) The sum of the first 𝑛 terms of the series is −3 ln 𝑥.
[6]
Find the value of 𝑛.
HL_Arithmetic, Geometric, Exponents, and Logs [21 marks]
QUESTION 1
1
Show that log𝑟 2 𝑥 = 2 log𝑟 𝑥 where 𝑟, 𝑥 ∈ ℝ+ .
[2]
QUESTION 2
Find the solution of log2 𝑥 − log2 5 = 2 + log2 3.
[4]
QUESTION 3
Solve the equation log2 (𝑥 + 3) + log2 (𝑥 − 3) = 4.
[5]
QUESTION 4
The 1st, 4th and 8th terms of an arithmetic sequence, with common difference 𝑑, 𝑑 ≠ 0, are the first
three terms of a geometric sequence, with common ratio 𝑟. Given that the 1st term of both sequences
is 9 find
(a) the value of 𝑑;
[4]
(b) the value of 𝑟;
[1]
© International Baccalaureate Organization, 2023