Question

Question 11 A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Class X1 - Maths -Sequences and Series Page 199

Answer 1

Let the GP is a , ar , ar² , ar³, ......... ar²ⁿ⁻² , ar²ⁿ⁻¹
we see that, a , ar² , ar⁴ , ar⁶ ......... occupy in odd places and ar , ar³ , ar⁵ , ar⁷ ...........occupy in places .
a/c to question,
sum of all terms = 5 × sum of terms occupying off places

e.g. a + ar + ar² + ........ + ar²ⁿ⁻¹ = 5 × (a , ar² , ar⁴ , ar⁶ .........)

a(r²ⁿ - 1)/(r - 1) = 5 × a{(r²)ⁿ - 1}/(r² - 1)
(r²ⁿ - 1)/(r - 1) = 5 × (r²ⁿ - 1)/(r - 1)(r + 1)
1 = 5/(r + 1)
1 + r = 5
r = 4

hence, common ratio = 4