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  a₁₀ −a₅ = 200

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`

We are given,  `a_10 - a_5 = 200`

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`

Here we find `a_30` and  `a_20`

So, for 10^th term,

`a_10 = a + (10 - 1)d`

= a + (9)d

Also for 5 th term

`a_5 = a + (5 - 1)d`

= a + (4)c

So

`a_10 - a_5 = (a +_ 9d)- (a + 4d)`

200 = a + 9d - a - 4d

200 = 5d

`d = 200/5`

d = 40

Therefore the common difference for the A.P. is d = 40