QJEEM21Ma3820: JEE Main 2021 Mathematics - Class 11 JEEM21Ma3820 Solution: In an increasing geometric series, the sum of the second and…

Question

Verified Solution • 2025–26 Session moderate

In an increasing geometric series, the sum of the second and the sixth term is

{{25} \over 2}

and the product of the third and fifth term is 25. Then, the sum of 4^th, 6^th and 8^th terms is equal to :

Accepted Answer

a, ar, ar², .....

{T_2} + {T_6} = {{25} \over 2} \Rightarrow ar(1 + {r^4}) = {{25} \over 2}

{a^2}{r^2}{(1 + {r^4})^2} = {{625} \over 4}

.... (1)

{T_3}.{T_5} = 25 \Rightarrow (a{r^2})(a{r^4}) = 25

{a^2}{r^6} = 25

.....(2)

On dividing (1) by (2)

{{{{(1 + {r^4})}^2}} \over {{r^4}}} = {{25} \over 4}

4{r^8} - 14{r^4} + 4 = 0

(4{r^4} - 1)({r^4} - 4) = 0

{r^4} = {1 \over 4},4 \Rightarrow {r^4} = 4

(an increasing geometric series)

{a^2}{r^6} = 25 \Rightarrow {(a{r^3})^2} = 25

{T_4} + {T_6} + {T_8} = a{r^3} + a{r^5} + a{r^7}

= a{r^3}(1 + {r^2} + {r^4})

= 5(1 + 2 + 4) = 35

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