The sum of the first ten terms of an A.P. is | vedclass

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

Question

(A) Let the $A$.$P$. be $a, a+d, \dots$ and the $G$.$P$. be $g, gr, \dots$.Given that the sum of the first ten terms of the $A$.$P$. is $160$, we have

Vedclass · Accepted Answer

$\frac{34}{9}$

Vedclass · Answer

(A) Let the $A$.$P$. be $a, a+d, \dots$ and the $G$.$P$. be $g, gr, \dots$.Given that the sum of the first ten terms of the $A$.$P$. is $160$, we have $S_{10} = \frac{10}{2}(2a + 9d) = 160$, which simplifies to $2a + 9d = 32$.Given that the sum of the first two terms of the $G$.$P$. is $8$, we have $g + gr = 8$, or $g(1+r) = 8$.We are given $a = r$ and $g = d$. Substituting these into the equations:$2r + 9g = 32$ and $g(1+r) = 8$.From the second equation, $g = \frac{8}{1+r}$. Substituting this into the first equation:$2r + 9(\frac{8}{1+r}) = 32 \Rightarrow 2r(1+r) + 72 = 32(1+r)$.$2r^2 + 2r + 72 = 32 + 32r \Rightarrow 2r^2 - 30r + 40 = 0 \Rightarrow r^2 - 15r + 20 = 0$.Let the roots be $r_1$ and $r_2$. Then $r_1 + r_2 = 15$ and $r_1r_2 = 20$.The first term of the $G$.$P$. is $g = \frac{8}{1+r}$. The sum of all possible values of $g$ is:$g_1 + g_2 = \frac{8}{1+r_1} + \frac{8}{1+r_2} = 8 \left( \frac{1+r_2 + 1+r_1}{(1+r_1)(1+r_2)} \right) = 8 \left( \frac{2 + (r_1+r_2)}{1 + (r_1+r_2) + r_1r_2} \right)$.Substituting the values: $8 \left( \frac{2 + 15}{1 + 15 + 20} \right) = 8 \left( \frac{17}{36} \right) = \frac{34}{9}$.

Similar Questions

Let $I(n) = n^n$ and $J(n) = 1 \times 3 \times 5 \times \ldots \times (2n - 1)$ for all $n > 1, n \in N$. Then:

If $a$ is the arithmetic mean of $b$ and $c$ and $G_1, G_2$ are the two geometric means between them,then $G_1^3 + G_2^3 = $

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. The greatest value of $n$ for which $a_n$ is the greatest is:

If $a, b, c$ are in $GP$ and $4a, 5b, 4c$ are in $AP$ such that $a + b + c = 70$,then the value of $a^3 + b^3 + c^3$ is

If the $(m + 1)^{th}$,$(n + 1)^{th}$ and $(r + 1)^{th}$ terms of an $A.P.$ are in $G.P.$ and $m, n, r$ are in $H.P.$,then the value of the ratio of the common difference to the first term of the $A.P.$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module