The sum of the 3^{rd} and the 4^{th} terms of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the first three terms of the $G.P.$ be $a, ar, ar^2$.According to the problem,the product of the first three terms is $1000$:$a(ar)(ar^2) = 10

Vedclass · Accepted Answer

$320$

Vedclass · Answer

(D) Let the first three terms of the $G.P.$ be $a, ar, ar^2$.According to the problem,the product of the first three terms is $1000$:$a(ar)(ar^2) = 1000$ $\Rightarrow (ar)^3 = 1000$ $\Rightarrow ar = 10$.The sum of the $3^{rd}$ and $4^{th}$ terms is $60$:$ar^2 + ar^3 = 60 \Rightarrow ar(r + r^2) = 60$.Substituting $ar = 10$ into the equation:$10(r + r^2) = 60 \Rightarrow r^2 + r - 6 = 0$.Factoring the quadratic equation:$(r + 3)(r - 2) = 0 \Rightarrow r = 2$ or $r = -3$.Case $1$: If $r = 2$,then $a(2) = 10 \Rightarrow a = 5$.Case $2$: If $r = -3$,then $a(-3) = 10 \Rightarrow a = -10/3$.Since the first term $a$ must be positive,we choose $a = 5$ and $r = 2$.The $7^{th}$ term is given by $T_7 = ar^6 = 5(2)^6 = 5 	imes 64 = 320$.

The sum of the $3^{rd}$ and the $4^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000$. If the first term of this $G.P.$ is positive,then its $7^{th}$ term is

Similar Questions

If the sum of an infinite $GP$ is $9$ and the sum of the first two terms is $5$,then their common ratio is .....

The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

If $x_1, x_2, x_3$ as well as $y_1, y_2, y_3$ are in geometric progression with the same common ratio,then the points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ are

$A$ person has $2$ parents,$4$ grandparents,$8$ great-grandparents,and so on. Find the total number of his ancestors during the $10$ generations preceding his own.

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

Vedclass Test Series

Exam Paper Generator

Online Exam Module