# NCERT Class 11-Math's: Chapter –9 Sequence and Series Part 4

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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* > 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