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QJEEM21Ma262f: JEE Main 2021 Mathematics - Class 11 JEEM21Ma262f Solution: If the sum of an infinite GP a, ar, ar 2 , ar 3 , ....... is 15 …

If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ....... is :

Multiple Choice Options

  • A 52{5 \over 2}
  • B 12{1 \over 2}
  • C 252{25 \over 2}
  • D 92{9 \over 2}

Step-by-Step Verified Solution

Sum of infinite terms :

a1−r=15{a \over {1 - r}} = 15 ..... (i)

Series formed by square of terms :

a2, a2r2, a2r4, a2r6 .......

Sum = a21−r2=150{{{a^2}} \over {1 - {r^2}}} = 150

⇒a1−r.a1+r=150⇒15.a1+r=150\Rightarrow {a \over {1 - r}}.{a \over {1 + r}} = 150 \Rightarrow 15.{a \over {1 + r}} = 150

⇒a1+r=10\Rightarrow {a \over {1 + r}} = 10 ...... (ii)

by (i) and (ii), a = 12; r = 15{1 \over 5}

Now, series : ar2, ar4, ar6

Sum = ar21−r2=12.(125)1−125=12{{a{r^2}} \over {1 - {r^2}}} = {{12.\left( {{1 \over {25}}} \right)} \over {1 - {1 \over {25}}}} = {1 \over 2}

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