# If a, b, c, d are in G.P, prove that (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in G.P

**Solution:**

It is given that a, b, c, d are in G.P.

Therefore,

b^{2} = ac ....(1)

c^{2} = bd ....(2)

ad = bc ....(3)

We need to prove (a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d ^{n}) are in G.P. i.e., we have to prove that

(b^{n} + c^{n)2} = (a^{n} + b^{n})×(c^{n} + d^{n})

Consider,

(b^{n} + c^{n})^{2} = b^{2}^{n} + 2b^{n}c^{n} + c^{2}^{n}

= (b^{2})^{n} + 2b^{n}c^{n} + (c^{2})^{n}

= (ac)^{n} + 2b^{n}c^{n} + (bd)^{n} [Using (1) and (2)]

= a^{n}c^{n} + b^{n}c^{n} + b^{n}c^{n} + b^{n}d^{n}

= a^{n}c^{n} + b^{n}c^{n} + a^{n}d^{n} + b^{n}d^{n} [Using (3)]

= c^{n} (a^{n} + b^{n}) + d^{n} (a^{n} + b^{n})

= (a^{n} + b^{n})(c^{n} + d ^{n})

Hence, (b^{n} + c^{n})^{2} = (a^{n} + b^{n})×(c^{n} + d^{n})

Thus, (a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d^{n}) are in G.P

NCERT Solutions Class 11 Maths Chapter 9 Exercise ME Question 17

## If a, b, c, d are in G.P, prove that (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in G.P

**Summary:**

If, a, b, c, d are in G.P. we proved that (a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d ^{n}) are in G.P