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

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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 > 2c … (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