# Sequences » Arithmetic progression

In an arithmetic sequence the difference between adjacent terms is always the same.
For an arithmetic sequence with a difference d and initial term t0 is:

The direct formula: tn = t0 + dn.

The recursive formula: tn = tn – 1 + d.

The sum of an arithmetic sequence can be calculated with:

sum = 12 · number of terms · (first term + last term) = 12(n + 1)(t0 + tn)

Example
Given is the following sequence: 43 + 58 + 73 + ... + 358.

 a. Give the direct formula. b. Give the recursive formula. c. Calculate the sum.

Answers

a.  For the direct formula t0 = 43 and d = 15. The direct formula is: tn = 43 + 15n.
b. The recursive formula for this t0 and d is: tn = tn – 1 + 15.
c. To calculate the sum, we first need to know how many terms are in the sequence.
The number of terms in the sequence is equal to the number of the last term +1, as you start counting at t0.
To calculate the number of the last term, we need to solve the following equation:
 43 + 15n = 358 15n = 315 n = 21
The number of the last term is 21, so the total number of terms is 22.

The sum of this sequence is then:
sum = 12 · 22 · (43 + 358) = 4411