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

20^{th} term is 10 more than the 18^{th} term.

#### Solution

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`

So for the nth term

Similarly for kth 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, the 20^{th} term is 10 more than the 18^{th} term. So, let us first find the 20^{th} term and 18^{th} term of the A.P.

Here

Let us take the first term as *a* and the common difference as *d*

Now, as we know,

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

So for 20th term (n = 120)

`a_20 = a + (20 - 1)d`

= a + 19d

Also for 18th term (n = 18)

`a_18 = a + (18 - 1)d`

= a + 17d

Now, we are given,

`a_20 = a_18 + 10`

On substituting the values, we get,

a + 19d = a + 17d + 10

19d - 17d = 10

2d = 10

`d = 10/2`

d = 5

Therefore, the common difference for the A.P. is d = 5