Question

Question 17 If a, b, c, d are in G.P, prove that (a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.

Class X1 - Maths -Sequences and Series Page 199

Answer 1

a , b , c , d are in GP .
Let b = ar , c = ar² , d = ar³

Now,
(aⁿ + bⁿ) = { aⁿ + (ar)ⁿ} = {aⁿ + aⁿrⁿ}
= aⁿ ( 1 + rⁿ )

(bⁿ + cⁿ) = {(ar)ⁿ + (ar²)ⁿ} = {aⁿrⁿ + aⁿr²ⁿ}
= aⁿrⁿ ( 1 + rⁿ)

(cⁿ + dⁿ) = {(ar²)ⁿ + (ar³)ⁿ} = {aⁿr²ⁿ + aⁿr³ⁿ}
= aⁿr²ⁿ ( 1 + rⁿ)

we observed that ,
(aⁿ + bⁿ) × (cⁿ + dⁿ) = a²ⁿr²ⁿ(1 + rⁿ)²
= {aⁿrⁿ (1 + rⁿ)}²
= { (bⁿ + cⁿ)}²
we know, that three terms A , B , C are in GP
if B² = AC
for this concept we see
(aⁿ + bⁿ) , ( bⁿ + cⁿ) , (cⁿ + dⁿ) are in GP