Arithmetic Sequence Calculator
nth term and sum, with the off-by-one that catches everyone.
Work out Arithmetic Sequence. nth term and sum, with the off-by-one that catches everyone. Says what the result does not mean.
Term 10
48
sum of the first 10 terms is 255
The nth term is a + (n−1)d — note the n−1, since the first term has had no difference applied yet. Off-by-one there is the commonest error in the topic. The sum is n/2 × (first + last), which is Gauss's trick: pair the first term with the last, the second with the second-last, and every pair has the same total. That is how he supposedly summed 1 to 100 in seconds — fifty pairs of 101. The average term always equals the average of the first and last, which is a quick way to check a sum by eye.
How the Arithmetic Sequence Calculator works
Enter a first term, a common difference and a term count for the nth term, the sum and the opening terms. The nth term uses n−1 differences rather than n, which is the single commonest slip in the topic.
Also known as: nth term calculator · arithmetic series sum calculator · common difference calculator · sum of an arithmetic sequence
Frequently asked questions
What is the formula for the nth term?
a + (n−1)d. The n−1 matters: the first term has had no difference applied to it yet, so the tenth term has had nine added, not ten.
How do I sum an arithmetic series?
n/2 × (first + last), or equivalently n/2 × (2a + (n−1)d). Pair the first term with the last, the second with the second-last, and every pair has the same total.
How did Gauss sum 1 to 100 so quickly?
By pairing. 1+100, 2+99, 3+98 — fifty pairs each totalling 101, giving 5,050. That pairing is exactly what the sum formula encodes.
How do I find the common difference?
Subtract any term from the one after it. If the difference is not the same for every consecutive pair, the sequence is not arithmetic.
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