How it works
In an arithmetic sequence each term is the previous one plus a common difference d. The nth term is
aₙ = a₁ + (n − 1)d
The (n − 1) is the most common mistake: to reach term 15 you add d only 14 times. If you know two terms, the difference is (change in value) ÷ (change in position).
The sum of an arithmetic series is the average of the first and last terms times the number of terms:
S = (first + last) ÷ 2 × (number of terms)
That's Gauss's trick: pair the first with the last, the second with the second-to-last, and so on. Counting terms: from a to b inclusive there are b − a + 1 integers.
Arithmetic Series practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Arithmetic sequence starts 7, 15, .... What is term 15?
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Answer: 119
d = 8, so term 15 = 7 + 14 × 8 = 7 + 112 = 119.
An arithmetic sequence has 1st term 14 and 5th term 46. What is the 20th term?
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Answer: 166
From term 1 to term 5 is 4 steps, so d = (46 − 14) ÷ 4 = 8. Term 20 = 14 + 19 × 8 = 166.
Find the sum of the first 20 positive integers.
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Answer: 210
(1 + 20) ÷ 2 × 20 = 21 × 10 = 210.
What is the sum of all integers from 11 to 119?
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Answer: 7085
There are 119 − 11 + 1 = 109 terms. Sum = (11 + 119) ÷ 2 × 109 = 65 × 109 = 7085.
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