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Arithmetic Series Practice Problems

Count the terms, find the first and last, then use S_n = n(a_1 + a_n)/2. The later problems run it backwards and put it in context.

  1. 1.

    Find .

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    .

  2. 2.

    Find the sum of the first 12 terms of

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    , so .

  3. 3.

    Find .

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    There are 20 terms, so .

  4. 4.

    Find the sum of the first 40 odd numbers.

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    The first odd numbers add to : .

  5. 5.

    Find .

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    gives , so .

  6. 6.

    An arithmetic sequence has and . Find the sum of its first 25 terms.

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    , so .

  7. 7.

    Evaluate .

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    First term 5, last term 63, 30 terms: .

  8. 8.

    Find the sum of all multiples of 7 from 7 to 98.

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    There are 14 multiples: .

  9. 9.

    How many terms of add up to 165?

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    simplifies to , so .

  10. 10.

    A theater has 18 seats in the first row and 2 more in each row after it. How many seats are in 20 rows?

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    The last row has seats, so the total is .

  11. 11.

    You save $25 in week 1 and $5 more each week after that. How much have you saved after 10 weeks, in dollars?

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    Week 10 is , so .

  12. 12.

    Find the sum of the first 100 even numbers.

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    The first even numbers add to : .
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MPhil in Mathematics, Founder of Mathovia
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Arithmetic Series Practice Problems

These problems follow the arithmetic series lesson. The sum is always the number of terms times the average of the first and last.

The formula

Common mistakes to watch for

  • Using the sequence formula instead of the sum. gives one term; adds them all.
  • Miscounting the terms. From to with difference , . Forgetting the is the classic slip.
  • Mixing up the two sum formulas. Use when you know the last term, and when you know .
  • Ignoring a negative difference. A decreasing series has ; keep the sign when finding later terms.

Frequently Asked Questions

What is the formula for the sum of an arithmetic series?+

S_n = n(a_1 + a_n)/2, or S_n = (n/2)[2a_1 + (n − 1)d] if you don’t know the last term.

How do I count the terms between two values?+

Solve a_n = a_1 + (n − 1)d for n. From 2 to 59 in steps of 3 is 20 terms.

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

Arithmetic Sequence Calculator

More practice problems

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