Exams Joint Entrance Examination Mathematics Sequences and Series Question 240 Question 240 of 519 Medium MCQ
In an increasing geometric series, the sum of the second and sixth term is 25 2 \dfrac{25}{2} 2 25 , and the product of the third and fifth term is 25 25 25 . Then the sum of the 4th, 6th and 8th terms is equal to:
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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 {a^2}{r^6} = 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^8} - 14{r^4} + 4 = 0 4 r 8 − 14 r 4 + 4 = 0
( 4 r 4 − 1 ) ( r 4 − 4 ) = 0 (4{r^4} - 1)({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 ) = a{r^3}(1 + {r^2} + {r^4}) = a r 3 ( 1 + r 2 + r 4 )
= 5 ( 1 + 2 + 4 ) = 35 = 5(1 + 2 + 4) = 35 = 5 ( 1 + 2 + 4 ) = 35 Answer the question to unlock