Question

The general term of a sequence is given by a_n = −4n + 15. Is the sequence an A.P.? If so, find its 15th term and the common difference.

Answer 1

In the given problem, we need to find that the given sequence is an A.P or not and then find its 15^th term and the common difference.

Here

`a_n = -4n + 15`

Now, to find that it is an A.P or not, we will find its few terms by substituting n = 1,2,3

So,

Substituting n = 1, we get

`a_1 = -4(1) + 15`

`a_1 = 11`

Substituting n = 2, we get

`a_2 = -4(2) + 15`

`a_2 = 7`

Substituting n = 3, we get

`a_3 = -4(3) + 15`

`a_3 = 3`

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

We find the common difference (d) = `a_2 - a_1 = a_3 - a_2`

Thus,

`a_2 - a_1 = 7 - 11`

= -4

Also

`a_3 - a_2 = 3 - 7`

= -4

Since `a_2 - a_1 = a_3 - a_2`

Hence, the given sequence is an A.P and its common difference is d = -4

Now to find its 15th using the formula `a_n = a + (n - 1)d`

First term (a) = 11

n = 15

Common difference (d) = −4

Substituting the above values in the formula

`a_15 = 11 + (15 - 1)(-4)`

`a_15 = 11 + (-56)`

`a_15 = -45`

Therefore `a_15 = -45`