Geometric Sequences

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A geometric sequence is a list of numbers where you multiply by the same number each time to get the next term. That number is the common ratio, r. For example, 2, 6, 18, 54, … multiplies by 3. The formula a_n = a_1 \cdot r^{n-1} finds any term.

Before this lesson: You should know arithmetic sequences, exponential functions, and zero and negative exponents.

What Is a Geometric Sequence?

A rumor starts with 2 people. Each person tells 3 new people the next day. So 2 people know on day 1, then 6, then 18, then 54. Each day’s number is 3 times the day before.

That is a geometric sequence. An arithmetic sequence adds the same number each time. A geometric sequence multiplies by the same number each time.

To find the common ratio, divide any term by the term right before it. Check that you get the same answer for every pair.

The sequence 2, 6, 18, 54, 162 shown in five boxes labeled a₁ through a₅. A curved arrow labeled times 3 joins each box to the next one.
Each term is 3 times the one before it, so the common ratio is r = 3.

Geometric Sequence Formulas

Explicit Formula

a_n = a_1 \cdot r^{\,n-1}

Here a_1 is the first term, r is the common ratio, and n is the term number. To reach the nth term, you multiply by r a total of n - 1 times. That is why the exponent is n - 1, not n.

Recursive Formula

a_1 = \text{first term}, \quad a_n = a_{n-1} \cdot r

This says each term is the term before it times r. Like any recursive formula, it must also give the first term.

How to Write the Explicit Formula

  1. Find the first term, a_1.
  2. Find r. Divide each term by the term before it. Check that the answer is the same every time.
  3. Put a_1 and r into a_n = a_1 \cdot r^{n-1}.
  4. Do not multiply a_1 and r together. The exponent goes on r only.
  5. Check your formula with n = 1 and n = 2.

Worked Examples

Example 1: Is It Geometric?

Decide if each sequence is geometric. If it is, find the common ratio.

  • a) 3, 6, 12, 24, …
  • b) 2, 5, 8, 11, …
  • c) 81, −27, 9, −3, …
  • d) 1, 4, 9, 16, …

a) 6 \div 3 = 2, 12 \div 6 = 2, 24 \div 12 = 2. Yes, r = 2.

b) 5 \div 2 = 2.5, but 8 \div 5 = 1.6. Not geometric. It adds 3 each time, so it is arithmetic.

c) -27 \div 81 = -\tfrac{1}{3}, and every other pair also gives -\tfrac{1}{3}. Yes, r = -\tfrac{1}{3}. A negative ratio makes the signs switch back and forth.

d) 4 \div 1 = 4, but 9 \div 4 = 2.25. Not geometric. These are the square numbers.

Example 2: Write the Formula and Find a Term

Write an explicit formula for 5, 15, 45, … Then find the 8th term.

Step 1: a_1 = 5 and r = 15 \div 5 = 3.

Step 2: a_n = 5 \cdot 3^{n-1}. Check: a_1 = 5 \cdot 3^0 = 5 and a_2 = 5 \cdot 3^1 = 15.

Step 3: a_8 = 5 \cdot 3^7 = 5 \cdot 2187 = 10{,}935.

Answer: a_n = 5 \cdot 3^{n-1}, and a_8 = 10{,}935.

Example 3: A Ratio That Is a Fraction

Write a recursive formula and an explicit formula for 64, 32, 16, 8, … Then find the 7th term.

Step 1: r = 32 \div 64 = \tfrac{1}{2}. The terms get smaller, because the ratio is between 0 and 1.

Step 2: Recursive: a_1 = 64, a_n = a_{n-1} \cdot \tfrac{1}{2}. Explicit: a_n = 64\left(\tfrac{1}{2}\right)^{n-1}.

Step 3: a_7 = 64\left(\tfrac{1}{2}\right)^6 = 64 \cdot \tfrac{1}{64} = 1.

Answer: a_7 = 1

Example 4: Find the Term Number

The sequence 2, 6, 18, … keeps going. Which term is 1,458?

Step 1: a_1 = 2 and r = 3, so a_n = 2 \cdot 3^{n-1}.

Step 2: Set it equal to 1,458 and divide by 2: 3^{n-1} = 729.

Step 3: Find the power of 3 that makes 729: 3^6 = 729. So n - 1 = 6 and n = 7.

Answer: 1,458 is the 7th term. Check: 2 \cdot 3^6 = 2 \cdot 729 = 1458.

Example 5: Word Problem

A ball is dropped from 81 feet. Each bounce reaches 2/3 of the height of the bounce before. The first bounce goes up 54 feet. How high is the 4th bounce?

Step 1: The bounce heights form a geometric sequence with a_1 = 54 and r = \tfrac{2}{3}. Check: 81 \cdot \tfrac{2}{3} = 54.

Step 2: a_n = 54\left(\tfrac{2}{3}\right)^{n-1}.

Step 3: a_4 = 54\left(\tfrac{2}{3}\right)^3 = 54 \cdot \tfrac{8}{27} = 16.

Answer: The 4th bounce is 16 feet high. The bounces are 54, 36, 24 and 16 feet.

Common Mistakes

MistakeExample of the errorFix
Using n instead of n - 1a_8 = 5 \cdot 3^8a_8 = 5 \cdot 3^7
Multiplying a_1 and r first5 \cdot 3^{n-1} = 15^{n-1}The exponent is on 3 only. Do the power first
Subtracting instead of dividing3, 6, 12: 6 - 3 = 3, so r = 3Divide: 6 \div 3 = 2, so r = 2
Dividing in the wrong order64, 32: 64 \div 32 = 2Later term ÷ earlier term: 32 \div 64 = \tfrac{1}{2}
Losing the negative sign81, −27: r = \tfrac{1}{3}-27 \div 81 = -\tfrac{1}{3}. The signs alternate

Geometric Sequences and Exponential Functions

A geometric sequence is an exponential function whose inputs are the term numbers 1, 2, 3, … For example, a_n = 2 \cdot 3^{n-1} gives the same outputs as f(n) = 2 \cdot 3^{n-1} for whole numbers. The graph is a set of separate points on an exponential curve.

Arithmetic sequenceGeometric sequence
Next termAdd dMultiply by r
Example3, 7, 11, 15, …3, 6, 12, 24, …
Explicit formulaa_n = a_1 + (n - 1)da_n = a_1 \cdot r^{n-1}
Recursive formulaa_n = a_{n-1} + da_n = a_{n-1} \cdot r
Same kind of function asLinearExponential

Many growth and decay problems are geometric sequences. A balance that earns compound interest once a year makes a geometric sequence with r = 1 + \text{rate}. Adding up the terms of a sequence is a series, which comes in Algebra 2.

Practice Problems

  1. Is 4, 12, 36, 108, … geometric? If so, find r.
  2. Is 10, 5, 5/2, 5/4, … geometric? If so, find r.
  3. Find the next two terms of 2, −8, 32, …
  4. Write an explicit formula for 7, 14, 28, 56, …
  5. Find a_6 for a_n = 3 \cdot 2^{n-1}.
  6. Write a recursive formula for 100, 20, 4, …
  7. A geometric sequence has a_1 = 4 and r = -3. Find a_5.
  8. A geometric sequence has a_2 = 12 and a_5 = 96. Find r and a_1.
  9. Which term of 3, 6, 12, … is 384?
  10. A tournament starts with 64 teams. Half the teams are knocked out in each round. Write a formula for the number of teams in round n. How many teams play in round 6?
Show answers
  1. Yes. 12 \div 4 = 3, 36 \div 12 = 3 and 108 \div 36 = 3, so r = 3.
  2. Yes. Each term is half the one before, so r = \tfrac{1}{2}.
  3. r = -8 \div 2 = -4. The next terms are 32(-4) = -128 and -128(-4) = 512.
  4. a_1 = 7 and r = 2, so a_n = 7 \cdot 2^{n-1}.
  5. a_6 = 3 \cdot 2^5 = 3 \cdot 32 = 96
  6. r = 20 \div 100 = \tfrac{1}{5}. a_1 = 100, a_n = a_{n-1} \cdot \tfrac{1}{5}.
  7. a_5 = 4(-3)^4 = 4 \cdot 81 = 324
  8. From a_2 to a_5 is 3 steps, so r^3 = 96 \div 12 = 8 and r = 2. Then a_1 = 12 \div 2 = 6.
  9. 3 \cdot 2^{n-1} = 384, so 2^{n-1} = 128 = 2^7. Then n - 1 = 7 and n = 8. It is the 8th term.
  10. a_n = 64\left(\tfrac{1}{2}\right)^{n-1}. Round 6: 64\left(\tfrac{1}{2}\right)^5 = 64 \cdot \tfrac{1}{32} = 2 teams (the final).

FAQ

What is the difference between an arithmetic and a geometric sequence?

An arithmetic sequence adds the same number each time. A geometric sequence multiplies by the same number each time. Check the differences for arithmetic and the ratios for geometric.

How do you find the common ratio?

Divide any term by the term right before it. For 5, 15, 45, … the ratio is 15 \div 5 = 3. Check at least two pairs.

Can the common ratio be negative or a fraction?

Yes. A negative ratio makes the signs switch, as in 81, −27, 9, −3. A ratio between 0 and 1 makes the terms shrink, as in 64, 32, 16. The ratio can’t be 0.

Is a geometric sequence an exponential function?

Yes. The formula a_n = a_1 \cdot r^{n-1} has the variable in the exponent. The only difference is that the inputs are the whole numbers 1, 2, 3, …

Summary

  • A geometric sequence multiplies by the same number, the common ratio r, each time.
  • Find r by dividing a term by the term before it.
  • Explicit formula: a_n = a_1 \cdot r^{n-1}. Recursive formula: a_n = a_{n-1} \cdot r, with a_1 given.
  • The exponent is n - 1, and it goes on r only.
  • A geometric sequence is an exponential function with inputs 1, 2, 3, …

Previous lesson: Compound Interest

Next lesson: Linear vs. Exponential Functions (coming soon)

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