# NCERT Class 11-Math՚S: Chapter – 9 Sequence and Series Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

Doorsteptutor material for IEO 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 10**:

If are in A. P. with common difference ; then the sum of the series is equal to

**Answer**:

We have

**Question 11**:

(i) If are four distinct positive quantities in A. P. , then show that

(ii) If are four distinct positive quantities in G. P. , then show that

**Answer**:

(i) Since are in A. P. , then A . , for the first three terms.

Therefore,

Squaring, we get

Similarly, for the last three terms

Multiplying (1) and (2) , we get

(ii) Since are in G. P.

Again . for the first three terms

Similarly, for the last three terms

➾ *b* + *d* > 2*c* … (4)

Adding (3) and (4) , we get

**Question 12**:

If are three consecutive terms of an A. P. and are three consecutive terms of a G. P. Then prove that

**Answer**:

We have as three consecutive terms of A. P. Then

Now

**Question 13**:

Find the natural number *a* for which where the function satisfies or all natural numbers and further .

**Answer**:

Given that

Therefore,

And so on. Continuing the process, we obtain

But, we are given