# NCERT Class 11-Math՚S: Chapter – 9 Sequence and Series Part 11 (For CBSE, ICSE, IAS, NET, NRA 2022)

Doorsteptutor material for IMO-Level-2 is prepared by world's top subject experts: Get full length tests using official NTA interface: all topics with exact weightage, real exam experience, detailed analytics, comparison and rankings, & questions with full solutions.

**Question 14**:

If are in A. P. , whose common difference is , show that

**Answer**:

We have to prove that

Consider LHS

Now we have to find value of d in terms of so that further simplification can be made

As are in AP having common difference as

Hence

Take sin on both sides

Substitute appropriate value of for each term in LHS

We know that

Hence proved

**Question 15**:

If the sum of *p* terms of an A. P. is *q* and the sum of *q* terms is *p*, show that the sum of terms is . Also, find the sum of first terms .

**Answer**:

The sum of n terms of an AP is given by

Where a is the first term and d is the common difference

Given that

Subtract (i) from (ii) that is (ii) – (i)

Using

We have to show that

Using (m) and (n)

Substitute d from (iii)

Now we have to find sum of terms that is

Using (m) and (n)

Substitute d from (iii)

Hence sun of terms is