Question

Show that the sequence defined by a_n = 5n −7 is an A.P, find its common difference.

Answer 1

In the given problem, we need to show that the given sequence is an A.P and then find its common difference.

Here

`a_n = 5n - 7`

Now, to show that it is an A.P, we will find its few terms by substituting n = 1,2,3,4,5

So,

Substituting n = 1, we get

`a_1 = 5(1) - 7`

`a_1 = -2`

Substituting n = 2, we get

`a_2 = 5(2) - 7`

`a_2 = 3`

Substituting n = 3, we get

`a_3 = 5(3) - 7`

`a_3 = 8`

Substituting n = 4, we get

`a_4 = 5(4) - 7`

`a_4 = 13`

Substituting n = 5, we get

`a_5 = 5(5) - 7`

`a_5 = 18`

Further, for the given sequence to be an A.P,

We find the common difference (d)

`a = a_2 - a_1 = a_3 - a_2`

Thus

`a_2 - a_1 = 3 - (-2)`

= 5

Also

`a_3 - a_2 = 8 - 3`

= 5

Since `a_2 - a_1 = a_3 - a_2`

Hence, the given sequence is an A.P and its common difference is d = 5