Common difference The sum of n terms of an arithmetic series

Common difference.

The sum of n terms of an arithmetic series is 5n^2-11n for all values of n. Detemine the common difference.

To determine the common difference, we\'ll have to consider to consecutive terms of the a.p.

We\'ll get:

an - an-1 = d

We know, from enunciation, that the sum of n terms of the a.p. is:

a1 + a2 + ... + an = 5n^2 - 11n

We\'ll determine an by subtracting both sides the sum: a1 + a2 + .... + an-1:

an = 5n^2 - 11n - (a1 + a2 + .... + an-1)

But a1 + a2 + .... + an-1 = 5(n-1)^2 - 11(n-1)

an = 5n^2 - 11n - 5(n-1)^2 + 11(n-1)

We\'ll expand the squares:

an = 5n^2 - 11n - 5n^2 + 10n - 5 + 11n - 11

We\'ll combine and eliminate like terms:

an = 10n - 16

Knowing the general term an, we can determine any term of the arithmetical series.

a1 = 10*1 - 16

a1 = 10 - 16

a1 = -6

a2 = 10*2 - 16

a2 = 20 - 16

a2 = 4

a3 = 10*3 - 16

a3 = 14

The common difference is the difference between 2 consecutive terms:

a2 - a1 = 4 + 6 = 10

d = 10

We can verify and we\'ll get a3 = a2 + d

14 = 4 + 10

14 = 14

So, the common difference of the given arithmetic series is d = 10.