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

(i) 11^th term is 5 and 13^th term is 79.

(ii) a₁₀ −a₅ = 200

(iii) 20^th term is 10 more than the 18^th term.

Answer 1

A.P: a, a+d, a+2d …

Here, we first need to write the expression for

Now, as we know,

So, for n^th term,

Similarly, for k^th term

So,

So,

(i) 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,

Now, we will find and using the formula

So,

Also,

Solving for a and d

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

Therefore, the common difference for the A.P. is.

(ii) We are given,

Here,

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

Now, as we know,

Here, we find a₃₀ and a₂₀.

So, for 10^th term,

Also, for 5^th term,

So,

Therefore, the common difference for the A.P. is.

(iii) 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,

So, for 20^th term (n = 20),

Also, for 18^th term (n = 18),

Now, we are given,

On substituting the values, we get,

Therefore, the common difference for the A.P. is