If the sum of the first 2n terms of the arith | vedclass

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

Question

(C) For the first arithmetic progression $2, 5, 8, \dots$,the first term $a_1 = 2$ and common difference $d_1 = 3$. The sum of the first $2n$ terms is

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(C) For the first arithmetic progression $2, 5, 8, \dots$,the first term $a_1 = 2$ and common difference $d_1 = 3$. The sum of the first $2n$ terms is given by:$S_{2n} = \frac{2n}{2} [2(2) + (2n - 1)3] = n(4 + 6n - 3) = n(6n + 1) \dots (1)$For the second arithmetic progression $57, 59, 61, \dots$,the first term $a_2 = 57$ and common difference $d_2 = 2$. The sum of the first $n$ terms is given by:$S'_n = \frac{n}{2} [2(57) + (n - 1)2] = \frac{n}{2} [114 + 2n - 2] = \frac{n}{2} [2n + 112] = n(n + 56) \dots (2)$Given that $S_{2n} = S'_n$,we have:$n(6n + 1) = n(n + 56)$Since $n 
eq 0$,we can divide by $n$:$6n + 1 = n + 56$$5n = 55$$n = 11$

If the sum of the first $2n$ terms of the arithmetic progression $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of the arithmetic progression $57, 59, 61, \dots$,then $n = \dots$

Similar Questions

If the sum of the $10$ terms of an $A.P.$ is $4$ times the sum of its $5$ terms,then the ratio of the first term to the common difference is:

The sum of all those terms of the arithmetic progression $3, 8, 13, \ldots, 373$ which are not divisible by $3$ is equal to $.......$.

Let the sum of the first $n$ terms of a non-constant $A.P.$,$a_1, a_2, a_3, \dots$ be $S_n = 50n + \frac{n(n - 7)}{2}A$,where $A$ is a constant. If $d$ is the common difference of this $A.P.$,then the ordered pair $(d, a_{50})$ is equal to

Let the sequence $a_1, a_2, a_3, \dots, a_{2n}$ form an $A.P.$ Then $a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2n - 1}^2 - a_{2n}^2 = $

What is the sum of $n$ arithmetic means between $a$ and $b$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module