Progression (Sequence and Series)Hard
Question
Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
Options
A.5
B.3/5
C.8/5
D.1/5
Solution
Let a = first term of G.P.
r = common ratio of G.P.; Then G.P. is a, ar, ar2
Given S∞ = 20 ⇒
= 20 ⇒ a = 20(1 - r) .....(i)
Also a2 + a2r2 + a2r4 + .... to ∞ = 100 ⇒
= 100 ⇒ a2 = 100(1 - r)(1 + r) .....(ii)
From (i), a2 = 400 (1-r)2 ; From (ii) and (iii), we get 100 (1 - r)(1+r) = 400 (1-r)2
⇒ 1 + r = 4 - 4r ⇒ 5r = 3 ⇒ r = 3/5
r = common ratio of G.P.; Then G.P. is a, ar, ar2
Given S∞ = 20 ⇒
= 20 ⇒ a = 20(1 - r) .....(i) Also a2 + a2r2 + a2r4 + .... to ∞ = 100 ⇒
= 100 ⇒ a2 = 100(1 - r)(1 + r) .....(ii) From (i), a2 = 400 (1-r)2 ; From (ii) and (iii), we get 100 (1 - r)(1+r) = 400 (1-r)2
⇒ 1 + r = 4 - 4r ⇒ 5r = 3 ⇒ r = 3/5
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