The sum of the first two terms of a G.P. is 3 | vedclass

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

Question

(B) Let $a$ be the first term and $r$ be the common ratio of the $G.P.$Given: $a_1 + a_2 = 36 \Rightarrow a + ar = 36$$\Rightarrow a(1 + r) = 36$ $...

Vedclass · Accepted Answer

$\frac{3280}{81}$

Vedclass · Answer

(B) Let $a$ be the first term and $r$ be the common ratio of the $G.P.$Given: $a_1 + a_2 = 36 \Rightarrow a + ar = 36$$\Rightarrow a(1 + r) = 36$ $...(1)$Also,$a_1 \cdot a_3 = 9 \cdot a_2 \Rightarrow a \cdot ar^2 = 9 \cdot ar$$\Rightarrow ar^2 = 9$ $...(2)$From $(1)$,$a = \frac{36}{1+r}$. Substituting this into $(2)$:$\frac{36}{1+r} \cdot r^2 = 9 \Rightarrow 4r^2 = 1 + r \Rightarrow 4r^2 - r - 1 = 0$Solving for $r$: $r = \frac{1 \pm \sqrt{1 - 4(4)(-1)}}{8} = \frac{1 \pm \sqrt{17}}{8}$.Wait,re-evaluating the condition: $a_1 a_3 = 9 a_2 \Rightarrow a(ar^2) = 9(ar) \Rightarrow a^2 r^2 = 9ar$. Since $a, r 
eq 0$,$ar = 9$.Substitute $ar = 9$ into $a + ar = 36$: $a + 9 = 36 \Rightarrow a = 27$.Then $27r = 9 \Rightarrow r = \frac{1}{3}$.The sum of the first $n$ terms is $S_n = \frac{a(1-r^n)}{1-r}$.$S_8 = \frac{27(1 - (1/3)^8)}{1 - 1/3} = \frac{27(1 - 1/6561)}{2/3} = \frac{81}{2} \cdot \frac{6560}{6561} = \frac{3280}{81}$.

The sum of the first two terms of a $G.P.$ is $36$ and the product of the first and the third terms is $9$ times the second term,then find the sum of the first $8$ terms.

Similar Questions

If $\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3 = \frac{k}{21}$,then $k$ equals

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an arithmetic sequence are $a$,$b$,and $c$ respectively,then the value of $[a(q - r) + b(r - p) + c(p - q)]$ is:

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

If $\sum_{n = 1}^5 \frac{1}{n(n + 1)(n + 2)(n + 3)} = \frac{k}{3}$,then $k$ is equal to

Let $a, b$ and $c$ be in $G.P.$ with common ratio $r,$ where $a \ne 0$ and $0 < r \le \frac{1}{2}.$ If $3a, 7b$ and $15c$ are the first three terms of an $A.P.,$ then the $4^{th}$ term of this $A.P.$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module