# Find the value of n so that a^{n + 1} + b^{n + 1}/a^{n} + b^{n} may be the geometric mean between a and b

**Solution:**

It is known that G.M. of a and b is √ ab

By the given condition

a^{n + 1} + b^{n + 1}/a^{n} + b^{n} = √ab

By squaring both sides, we obtain

(a^{n + 1} + b^{n + 1})^{2}/(a^{n} + b^{n})^{2} = ab

⇒ a^{2n + 2} + 2a^{n + 1}b^{n + 1} + b^{2n + 2}

= (ab)(a^{2n} + 2a^{n}b^{n} + b^{2n})

⇒ a^{2n + 2} + 2a^{n + 1}b^{n + 1} + b^{2n + 2}

= a^{2n + 1}b + 2a^{n + 1}b^{n + 1} + ab^{2n + 1}

⇒ a^{2n + 2} + b^{2n + 2} = a^{2n + 1}b + ab^{2n + 1}

⇒ a^{2n + 2} - ab^{2n + 1} = ab^{2n + 1} - b^{2n + 2}

⇒ a^{2n}^{+1} (a - b) = b^{2n}^{+1} (a - b)

⇒ (a/b)^{2n}^{+1} = 1 = (a/b)^{0}

⇒ 2n + 1 = 0

⇒ n = - 1/2

Thus, the value of n = - 1/2

NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.3 Question 27

## Find the value of n so that a^{n + 1} + b^{n + 1}/a^{n} + b^{n} may be the geometric mean between a and b.

**Summary:**

We had to find the value of n so that the expression a^{n + 1} + b^{n + 1}/a^{n} + b^{n} may be the geometric mean between a and b

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