Geometric sequences and series — IB Maths AA practice questions

Question 1

The numbers $3$, $x$, $27$ are in geometric progression (with positive ratio). Determine $x$.

Accepted Answer

$x^2 = 3 \cdot 27 = 81 \Rightarrow x = 9$ (positive ratio).

Answer

$5$

Answer

$9$

Answer

$15$

Answer

$\sqrt{30}$

Question 2

Compute the sum of the first $10$ terms of the geometric progression $1, 2, 4, 8, \ldots$.

Accepted Answer

$S_{10} = 1 \cdot \dfrac{2^{10} - 1}{2 - 1} = 1023$.

Answer

$511$

Answer

$512$

Answer

$1023$

Answer

$1024$

Question 3

Compute $\displaystyle\sum_{k=0}^{5} 2^k$.

Accepted Answer

With $r=2$ and $n=5$: $\dfrac{2^6 - 1}{2-1} = \dfrac{63}{1} = 63$. (Check: $1+2+4+8+16+32 = 63$.)

Answer

$31$

Answer

$62$

Answer

$63$

Answer

$127$

Question 4

What is the sum of the infinite geometric series $1 + 	frac{1}{2} + 	frac{1}{4} + \cdots$?

Accepted Answer

Geometric series with $a=1$, $r=	frac{1}{2}$. Sum $= 	frac{a}{1-r} = 	frac{1}{1-1/2} = 2$.

Answer

$\dfrac{3}{2}$

Answer

$2$

Answer

$\dfrac{5}{2}$

Answer

Diverges

Question 5

In a geometric progression with $b_1 = 5$ and $q = 2$, the term $b_n = 320$. Determine $n$.

Accepted Answer

$5 \cdot 2^{n-1} = 320 \Rightarrow 2^{n-1} = 64 = 2^6 \Rightarrow n - 1 = 6 \Rightarrow n = 7$.

Answer

$5$

Answer

$6$

Answer

$7$

Answer

$8$

Question 6

Compute the sum $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \cdots$.

Accepted Answer

$b_1 = 1$, $q = \dfrac{1}{2}$: $S = \dfrac{1}{1 - 1/2} = 2$.

Answer

$\dfrac{3}{2}$

Answer

$2$

Answer

$\dfrac{5}{2}$

Answer

$\infty$

Question 7

A bank deposit of $\$1{,}000$ earns annual compound interest at $10\%$. After $3$ years, the balance is closest to:

Accepted Answer

$A = 1000 \cdot 1.1^3 = 1000 \cdot 1.331 = \$1{,}331$.

Answer

$\$1{,}300$

Answer

$\$1{,}320$

Answer

$\$1{,}331$

Answer

$\$1{,}350$

Question 8

In the geometric progression $(b_n)_{n \ge 1}$, $b_1 = 2$ and $q = 3$. Determine $b_5$.

Accepted Answer

$b_5 = b_1 \cdot q^{4} = 2 \cdot 3^{4} = 2 \cdot 81 = 162$.

Answer

$54$

Answer

$81$

Answer

$162$

Answer

$486$

Question 9

The numbers $4, 12, 36, 108, \ldots$ form a geometric progression. The common ratio $q$ equals:

Accepted Answer

$q = \dfrac{b_2}{b_1} = \dfrac{12}{4} = 3$.

Answer

$2$

Answer

$\dfrac{1}{3}$

Answer

$3$

Answer

$4$

Question 10

The numbers $a, b, c$ are in GP and $a + b + c = 14$, $abc = 64$. The middle term $b$ equals:

Accepted Answer

Since $b^2 = ac$, the product $abc = b \cdot ac = b \cdot b^2 = b^3 = 64$, so $b = 4$.

Answer

$2$

Answer

$3$

Answer

$4$

Answer

$6$

Geometric sequences and series practice questions — IB Maths AA

Problems & worked solutions

Problem 1 — Geometric sequences and series

Problem 2 — Geometric sequences and series

Problem 3 — Geometric sequences and series

Problem 4 — Geometric sequences and series

Problem 5 — Geometric sequences and series

Problem 6 — Geometric sequences and series

Problem 7 — Geometric sequences and series

Problem 8 — Geometric sequences and series

Problem 9 — Geometric sequences and series

Problem 10 — Geometric sequences and series