In a geometric sequence, $t_3 = 12$ and $t_6 = 324$. Find $a$, $r$ and $t_9$.

$a = \frac{4}{3}$, $r = 3$, $t_9 = 8748$

Grade 11 · MCR3U · Functions

MCR3U: Geometric Sequences and Series

A geometric sequence multiplies by the same number at every step, so it grows or shrinks faster and faster. As with arithmetic sequences, one formula gives any term and one adds up the terms, and the sum has a neat derivation worth knowing.

How It Works

Geometric sequences. Each term is the one before times a common ratio $r$ : $3, 6, 12, 24, \dots$ has $a = 3$ and $r = 2$ . To check a sequence, divide each term by the one before. A negative $r$ makes the signs alternate, and $r$ between $0$ and $1$ makes the terms shrink.

The general term. $t_n = ar^{n - 1}$ , since you multiply by $r$ only $n - 1$ times. As a recursive formula, $t_1 = a$ and $t_n = r \cdot t_{n - 1}$ .

How many terms. Set $ar^{n - 1}$ equal to the last term, divide by $a$ , and write both sides as powers of $r$ . For $3, 6, \dots, 768$ : $2^{n - 1} = 256 = 2^8$ , so $n - 1 = 8$ and $n = 9$ .

Geometric series. Write $S_n = a + ar + \dots + ar^{n - 1}$ , multiply both sides by $r$ , and subtract the first line from the second. Every term cancels except two, leaving $rS_n - S_n = ar^n - a$ . So $S_n = \frac{a(r^n - 1)}{r - 1}$ for any $r \ne 1$ . If you know the last term $t_n$ rather than $n$ , the same algebra gives $S_n = \frac{rt_n - a}{r - 1}$ .

Two terms given. Divide one general term by the other and $a$ cancels: $\frac{t_6}{t_3} = \frac{ar^5}{ar^2} = r^3$ . Find $r$ , then $a$ .

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
For the sequence $5, 15, 45, \dots$ find $t_8$ .
Show solution
1. $a = 5$ and $r = \frac{15}{5} = 3$ .
2. $t_8 = 5(3)^7 = 5(2187)$ .
Answer: $t_8 = 10935$
Easy
Find the sum of the first $10$ terms of $2 + 6 + 18 + \dots$
Show solution
1. $a = 2$ , $r = 3$ and $n = 10$ .
2. $S_{10} = \frac{2(3^{10} - 1)}{3 - 1} = 3^{10} - 1 = 59049 - 1$ .
Answer: $S_{10} = 59048$
Medium
In a geometric sequence, $t_3 = 12$ and $t_6 = 324$ . Find $a$ , $r$ and $t_9$ .
Show solution
1. $\frac{t_6}{t_3} = r^3 = \frac{324}{12} = 27$ , so $r = 3$ .
2. $ar^2 = 12$ gives $9a = 12$ , so $a = \frac{4}{3}$ .
3. $t_9 = t_6 \cdot r^3 = 324(27) = 8748$ .
Answer: $a = \frac{4}{3}$ , $r = 3$ , $t_9 = 8748$
Medium
Find the sum $3 - 6 + 12 - 24 + \dots + 3072$ .
Show solution
1. $a = 3$ and $r = -2$ . Number of terms: $3(-2)^{n - 1} = 3072$ , so $(-2)^{n - 1} = 1024 = (-2)^{10}$ and $n = 11$ .
2. $S_{11} = \frac{3\left((-2)^{11} - 1\right)}{-2 - 1} = \frac{3(-2049)}{-3} = 2049$ .
3. Check with the last term: $\frac{rt_n - a}{r - 1} = \frac{-2(3072) - 3}{-3} = \frac{-6147}{-3} = 2049$ .
Answer: $2049$
Test level
A ball is dropped from $10$ m and each bounce reaches $80\%$ of the height of the one before. How far has it travelled up and down when it hits the ground for the 6th time?
Show solution
1. The first fall is $10$ m. After that, each bounce goes up and comes back down the same distance.
2. Between the 1st and 6th landing there are $5$ bounces, with heights $8, 6.4, 5.12, \dots$ : a geometric sequence with $a = 8$ and $r = 0.8$ .
3. $S_5 = \frac{8(1 - 0.8^5)}{1 - 0.8} = 40(1 - 0.32768) = 26.8928$ .
4. Total: $10 + 2(26.8928) = 63.7856$ .
Answer: About $63.79$ m

Learn It on Martian Lab

Martian Lab teaches this in the Geometric Sequences lesson of its MCR3U Discrete Functions unit. Watch it, then take the Discrete Functions practice tests to see if it stuck.

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Free Discrete Functions practice test, with solutions

More Discrete Functions topics: Arithmetic Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Exponent Laws and Rational Exponents · Graphing Exponential Functions · Exponential Growth and Decay · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Transformations of Sine and Cosine · Sinusoidal Word Problems

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