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Geometric Sequences Practice Problems

Find the common ratio, then use a_n = a_1 · r^(n − 1). The later problems work backwards from two terms and apply the formula to money and bacteria.

  1. 1.

    Find the common ratio of

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    Each term is double the one before: .

  2. 2.

    Find the next term of

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    The ratio is , so the next term is .

  3. 3.

    A geometric sequence has and . Find .

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    .

  4. 4.

    Write the explicit formula for in the form (use n).

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    and , so .

  5. 5.

    A geometric sequence has and . Find .

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    .

  6. 6.

    Find the common ratio of

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    . A negative ratio makes the signs alternate.

  7. 7.

    Find the 8th term of

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    .

  8. 8.

    In a geometric sequence, and . Find the common ratio .

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    Three steps multiply by , so .

  9. 9.

    For the same sequence (, ), find .

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    .

  10. 10.

    Which term of is 384?

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    gives , so .

  11. 11.

    A $20,000 car loses 15% of its value each year. What is it worth after 3 years, in dollars?

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    Each year multiplies the value by 0.85: .

  12. 12.

    A culture starts with 50 bacteria and doubles every hour. How many are there after 6 hours?

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    After 6 doublings: . (The count after hours is the term .)
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MPhil in Mathematics, Founder of Mathovia
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Geometric Sequences Practice Problems

These problems follow the geometric sequences lesson. In every one, each term is the previous term multiplied by the same number.

The formula

Common mistakes to watch for

  • Using instead of . The first term already counts, so .
  • Finding by subtracting. The common ratio is a quotient: , not .
  • Losing a negative ratio. If the terms alternate in sign, is negative; keep the brackets in .
  • Confusing it with an arithmetic sequence. Check whether consecutive terms share a ratio (geometric) or a difference (arithmetic).

Frequently Asked Questions

What is the formula for the nth term of a geometric sequence?+

a_n = a_1 · r^(n − 1), where a_1 is the first term and r is the common ratio.

How do I find the ratio from two terms that are not next to each other?+

Divide the later term by the earlier one, then take the root matching the number of steps. From a_2 = 12 to a_5 = 324, r³ = 27, so r = 3.

How do I type my answers?+

Type numbers without units or commas, and fractions like 7/9. For formulas, use n and ^ for powers, such as 3n - 1 or 5*2^(n - 1).

Calculators for this topic

Geometric Sequence Calculator

More practice problems

Arithmetic Sequences Practice Problems Geometric Series Practice Problems Exponential Growth and Decay Practice Problems