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Arithmetic vs Geometric Sequences Practice Problems

Check the differences, then the ratios. If the differences are constant it is arithmetic; if the ratios are constant it is geometric; sometimes it is neither, or both.

  1. 1.

    Is arithmetic, geometric, both or neither?

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    It adds 4 each time, so it is arithmetic.

  2. 2.

    Is arithmetic, geometric, both or neither?

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    It multiplies by 3 each time, so it is geometric.

  3. 3.

    Is arithmetic, geometric, both or neither?

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    The differences are 3, 5, 7 and the ratios 4, 2.25, 1.78. Neither is constant: these are the square numbers.

  4. 4.

    Is arithmetic, geometric, both or neither?

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    It subtracts 4 each time: arithmetic with .

  5. 5.

    Is arithmetic, geometric, both or neither?

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    It multiplies by each time: geometric.

  6. 6.

    Is arithmetic, geometric, both or neither?

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    It multiplies by each time: geometric, with alternating signs.

  7. 7.

    Is arithmetic, geometric, both or neither?

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    Each term is the sum of the two before it (the Fibonacci sequence). Neither the differences nor the ratios are constant.

  8. 8.

    Is arithmetic, geometric, both or neither?

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    It is arithmetic with and geometric with , so both.

  9. 9.

    Is the sequence arithmetic, geometric, both or neither?

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    A linear formula in gives a constant difference, here 4.

  10. 10.

    Is the sequence arithmetic, geometric, both or neither?

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    An exponential formula in gives a constant ratio, here 2.

  11. 11.

    The sequence is arithmetic. Find .

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    The middle term of an arithmetic sequence is the average of its neighbours: .

  12. 12.

    The sequence is geometric and is positive. Find .

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    The middle term of a geometric sequence is the geometric mean: .
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Arithmetic vs Geometric Sequences Practice Problems

These problems follow the arithmetic vs geometric sequences lesson. Type arithmetic, geometric, both or neither, or a number where one is asked for.

Quick test

CheckConstant?The sequence is
yesarithmetic
yesgeometric
bothyesboth (a constant sequence)

Common mistakes to watch for

  • Checking only the first two terms. Confirm the difference or ratio holds for every consecutive pair.
  • Ignoring sequences that are neither. has no common difference or ratio.
  • Mixing the formulas. Arithmetic: . Geometric: .
  • Forgetting a constant sequence is both. is arithmetic with and geometric with .

Frequently Asked Questions

How do you tell if a sequence is arithmetic or geometric?+

Subtract neighbouring terms. If the differences are all equal, it is arithmetic. If not, divide them. If the ratios are all equal, it is geometric.

Can a sequence be both arithmetic and geometric?+

Only a constant sequence such as 5, 5, 5, which has d = 0 and r = 1.

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

Sequence Calculator

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