16. Geometric Sequences · Mastering AMC 10/12

16.0a0%

Terms of a Geometric Sequence

In this section you will be able to spot a common ratio, jump straight to any term with $g_n = g_1 r^{n-1}$, and link two terms without finding $r$.

Two sequences share a ratio: one starts at 27, the other at 99, and term 15 of the first equals term 11 of the second. Find term 9 of the first. You never need the ratio itself.

In an arithmetic sequence you add the same number each time. In a geometric sequence you multiply by the same number each time: the common ratio $r$ .

To reach term $n$ from term 1, you multiply by $r$ exactly $n - 1$ times. So $g_n = g_1 r^{n-1}$ . The same idea links any two terms: $g_n = g_m r^{n-m}$ .

Look: the middle of three equally spaced terms is their geometric mean: $g_8^2 = g_5 \cdot g_{11}$ , because both sides equal $g_1^2 r^{14}$ .

Exploreplay first, formulas later

Try this: Step the terms n from 1 to 6. Each bar is how many times the one before? Read g₆ = 3 × 2⁵: why is the exponent 5, not 6?
Try this: Set the ratio to −2. What happens to the signs of the terms? Does the size still double?
Try this: Set the ratio to 1/2. The terms shrink, but by the same factor every time. That factor is still the common ratio.

Name it

Definition 16.0.1 · Geometric Sequence

A geometric sequence is a sequence of numbers with the same ratio between consecutive terms.

1, 2, 4, 8, 16, 32, \ldots, 1024

is a geometric sequence because there is always a ratio of $2$ between consecutive terms.

Definition 16.0.2 · Geometric Sequence Notation

In general, the terms of a geometric sequence can be represented as

g_1, g_2, g_3, g_4, \ldots, g_n

where

$r$ is the common ratio between consecutive terms
$n$ is the number of terms

Theorem 16.0.4 · nth Term in a Geometric Sequence

g_n = g_1 \cdot r^{n-1}

g_n = g_m \cdot r^{n-m}

Watch itpredict, then reveal

Example 16.1

AIME

Two geometric sequences $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ have the same common ratio, with $a_1 = 27$ , $b_1 = 99$ , and $a_{15} = b_{11}$ . Find $a_9$ .

1
Call the common ratio $r$ . Write $a_{15}$ and $b_{11}$ with the $n$ th-term formula.

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Try it

Problem 16.1.1

MATHCOUNTS

The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$ . What is the eighth term of the sequence?

Problem 16.1.2

MATHCOUNTS

The third and fourth terms of an arithmetic sequence are the first and second terms of a geometric sequence. If the first two terms of the arithmetic sequence are $5, 2$ , then what is the fourth term of the geometric sequence?

Problem 16.1.3

MATHCOUNTS

The numbers $a$ , $b$ , $c$ , and $d$ form a geometric sequence, in that order. If $b$ is three more than $a$ , and $c$ is nine more than $b$ , what is the value of $d$ ?

Contest cornerreal contest problems

Problem 16.2.1(additional)

AIME

Call a 3-digit number geometric if it has 3 distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

Problem 16.2.5(additional)

AMC 10

Positive integers $a$ , $b$ , and $2009$ , with $a < b < 2009$ , form a geometric sequence with an integer ratio. What is $a$ ?

Problem 16.2.6(additional)

AIME

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by $30$ . Find the sum of the four terms.

Checkpoint for section 16.0a

Get 2 of 3 right to master this section. One try each, no hints.

16.0b0%

Geometric Series

In this section you will be able to add up a geometric sequence in one line, and find the sum of infinitely many terms when $|r| < 1$.

An infinite series adds up to 2005. Square every term and the new series adds up to ten times as much. What was the common ratio?

To add $g_1 + g_1 r + \cdots + g_1 r^{n-1}$ , multiply the sum by $r$ and subtract. Almost every term cancels, and you get $S_n = g_1 \frac{r^n - 1}{r - 1}$ .

When $|r| < 1$ , the term $r^n$ shrinks toward 0. The running total then settles on $\frac{g_1}{1 - r}$ . When $|r| \ge 1$ , it never settles.

Look: squaring every term of a geometric series gives another geometric series, with first term $g_1^2$ and ratio $r^2$ .

Exploreplay first, formulas later

Try this: Set r = 1/2 and step n up. How far is the running total from 2 after each term? What happens to that gap?
Try this: Try r = 1 and r = −1. Why does the limit line vanish?
Try this: Turn on Square every term. The squares form a new geometric series: what is its ratio? What does it add up to?

Name it

Theorem 16.0.5 · Sum of All Terms in a Finite Geometric Sequence

For $r \ne 1$ ,

S_n = g_1 \cdot \frac{r^n - 1}{r - 1}

Theorem 16.0.6 · Sum of All Terms in an Infinite Geometric Sequence

For $-1 < r < 1$ ,

S_\infty = \frac{g_1}{1 - r}

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Example 16.2

AIME

For $-1 < r < 1$ , let $S(r)$ denote the sum of the geometric series

12 + 12r + 12r^2 + 12r^3 + \cdots.

Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a) = 2016$ . Find $S(a) + S(-a)$ .

1
Use the infinite-sum formula. What is $S(r)$ as a fraction?

Predict first:
Click a choice or type its letter (A–E).

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Example 16.3

AIME

An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$ .

1
Call the first term $a$ and the ratio $r$ . What equation does "sum 2005" give?

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Try it

Problem 16.2.4(additional)

AMC 10

The sum of an infinite geometric series is a positive number $S$ , and the second term in the series is $1$ . What is the smallest possible value of $S$ ?

Problem 16.2.8(additional)

Let $p$ and $q$ be real numbers with $|p| < 1$ and $|q| < 1$ such that

p + pq + pq^2 + pq^3 + \cdots = 2 \quad\text{and}\quad q + qp + qp^2 + qp^3 + \cdots = 3.

Find $100pq$ .

Contest cornerreal contest problems

Problem 16.2.2(additional)

AIME

The sum of the first $2011$ terms of a geometric sequence is $200$ . The sum of the first $4022$ terms is $380$ . Find the sum of the first $6033$ terms.

Problem 16.2.3(additional)

AIME

Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if

\frac{n}{810} = 0.d25d25d25\ldots

first term g₁9/25

ratio r1/100

terms n4

Terms: 9/25, 9/2500, 9/250000, ≈ 3.600e-7

Running total after each term

g₄ = 9/25 × (1/100)³ = ≈ 3.600e-7

g_n = g_1 \cdot r^{n-1}

S₄ = ≈ 0.3636

S_n = g_1 \cdot \frac{1 - r^n}{1 - r}

limit 4/11; gap left = ≈ 3.636e-9

S_\infty = \frac{g_1}{1-r} = \tfrac{4}{11}

Problem 16.2.10(additional)

AIMEChallenge

Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$ , and the second term of both series can be written in the form $\frac{\sqrt{m} - n}{p}$ , where $m$ , $n$ , and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m + 10n + p$ .

Checkpoint for section 16.0b

Get 2 of 3 right to master this section. One try each, no hints.

16.1

Practice Problems

In this section you will be able to mix arithmetic and geometric sequences in one problem.

Some problems hand you an arithmetic sequence and a geometric sequence that share terms. Write each condition as an equation, then let whole numbers narrow things down.

The other practice problems of 16.1 are the Try-it problems in 16.0a. Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 16.1.4

MATHCOUNTS

The positive integers $A$ , $B$ , and $C$ form an arithmetic sequence while the integers $B$ , $C$ , and $D$ form a geometric sequence. If $\frac{C}{B} = \frac53$ , what is the smallest possible value of $A + B + C + D$ ?

16.2

Additional Problems

In this section you will be able to use everything from this chapter on harder contest problems.

Two last challenges: a basketball game scored in sequences, and an AIME sequence that switches between geometric and arithmetic steps. Write out the first few terms and look for the pattern.

The other Additional Problems sit in sections 16.0a and 16.0b, next to the ideas they use.

Contest cornerreal contest problems

Problem 16.2.7(additional)

AMC 12

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?

Problem 16.2.9(additional)

AIMEChallenge

A sequence of positive integers with $a_1 = 1$ and $a_9 + a_{10} = 646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n \ge 1$ , the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ are in geometric progression, and the terms $a_{2n}, a_{2n+1}$ , and $a_{2n+2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than $1000$ . Find $n + a_n$ .