If the 2^{nd}, 5^{th}, and 9^{th} terms of a | vedclass

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

Question

(D) Let the first term of the $A.P.$ be $a$ and the common difference be $d$. The terms are given by $T_n = a + (n-1)d$.The $2^{nd}, 5^{th},$ and $9^{

Vedclass · Accepted Answer

$\frac{4}{3}$

Vedclass · Answer

(D) Let the first term of the $A.P.$ be $a$ and the common difference be $d$. The terms are given by $T_n = a + (n-1)d$.The $2^{nd}, 5^{th},$ and $9^{th}$ terms are $a+d, a+4d,$ and $a+8d$ respectively.Since these terms are in $G.P.$,the square of the middle term equals the product of the first and third terms:$(a+4d)^2 = (a+d)(a+8d)$Expanding both sides:$a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2$Subtracting $a^2$ and rearranging terms:$8d^2 = ad$Since the $A.P.$ is non-constant,$d 
eq 0$,so we can divide by $d$:$a = 8d$The terms of the $G.P.$ are:$T_2 = a+d = 8d+d = 9d$$T_5 = a+4d = 8d+4d = 12d$$T_9 = a+8d = 8d+8d = 16d$The common ratio $r$ is given by:$r = \frac{T_5}{T_2} = \frac{12d}{9d} = \frac{4}{3}$

If the $2^{nd}, 5^{th},$ and $9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$,then the common ratio of this $G.P.$ is:

Similar Questions

If $A.M.$ and $G.M.$ of two positive numbers $a$ and $b$ are $10$ and $8$ respectively,find the numbers.

Let $a_1, a_2, a_3$ be any positive real numbers,then which of the following statements is not true?

If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $G.P.$ such that $a < b < c$ and $a+b+c = \frac{3}{4}$,then the value of $a$ is

The $8^{\text{th}}$ common term of the series $S_1 = 3 + 7 + 11 + 15 + 19 + \dots$ and $S_2 = 1 + 6 + 11 + 16 + 21 + \dots$ is $.......$.

In a $G.P.$,the sum of three numbers is $14$. If $1$ is added to the first two numbers and subtracted from the third number,the series becomes an $A.P.$. Then,the greatest number is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module