Question

Consider the following arithmetic progression 6, 14, 22, 30, …, 1438 A. Find the number of terms B. Determine the sum using Gauss’s method.

Answer 1

There is a difference of 8 between consecutive terms:

6 = 6

6 + 8 = 14

6 + 8 + 8 = 22

6 + 8 + 8 + 8 = 30

and so on. The n'th term of the progression is then 6 + 8(n - 1). This means the last term is 180th in the sequence, since

1438 = 6 + 8(n - 1)  ==>  1432 = 8(n - 1)  ==>  179 = n - 1  ==>  n = 180

Let S be the sum of the series,

S = 6 + 14 + 22 + 30 + ... + 1438

Reversing the series, we have

S = 1438 + 1430 + 1422 + 1414 + ... + 6

Adding together terms in the same position gives

2S = (6 + 1438) + (14 + 1430) + (22 + 1422) + ... + (1438 + 6)

2S = 1444 + 1444 + 1444 + ... + 1444

We know there are 180 terms in the progression, so there are 180 copies of 1444 on the right side,

2S = 180 * 1444  ==>  S = (180 * 1444)/ 2

and so the sum is S = 129,960.