Question

Write the expression a_n- a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which

11^th term is 5 and the 13^th term is 79.

Answer 1

A.P: a, a+d, a+2d …

Here, we first need to write the expression for `a_n - a_k`

Now, as we know,

`a_n = a + (n - 1)d`

So for the nth term

`a_n = a + (n - 1)d`

Similarly for k^th term

`a_k = a + (k - 1)d`

So,

`a_n - a_k = (a  + nd - d) - (a + kd - d)`

= a + nd - d - a - kd + d

= nd - kd

= (n - k)d

So. `a_n - a_k = (n - k)d`

In the given problem, we are given 11^th and 13^th term of an A.P.

We need to find the common difference. Let us take the common difference as d and the first term as a.

Here,

`a_11 = 5`

`a_13 = 79`

Now we will find `a_11` and `a_13` using the formula `a_n = a + (n - 1)d`

So,

`a_11 = a + (11 - 1)d`

5 = a + 10d ......(1)

Also

`a_13 = a + (13 - 1)d`

5 = a + 10d ....(1)

Also

a_13 = a + (13 - 1)d

79 = a + 12d ....(2)

Solving for a and d

On subtracting (1) from (2), we get

79 - 5 = (a + 12d) - (a + 10d)

74 = a + 12d - a - 10d

74 = 2d

`d = 74/2`

d = 37

Therefore the common diffference for the A.P. is d = 37