If the first term and the last term of an ari | vedclass

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

Question

(D) Let the number of terms be $n$. The sum of an arithmetic progression is given by $S = \frac{n}{2}(a + \ell)$.From this,we can find $n = \frac{2S}{

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) Let the number of terms be $n$. The sum of an arithmetic progression is given by $S = \frac{n}{2}(a + \ell)$.From this,we can find $n = \frac{2S}{a + \ell}$.We also know the formula for the last term: $\ell = a + (n - 1)d$.Rearranging for $d$,we get $d = \frac{\ell - a}{n - 1}$.Substitute $n = \frac{2S}{a + \ell}$ into the equation for $d$:$d = \frac{\ell - a}{\frac{2S}{a + \ell} - 1} = \frac{\ell - a}{\frac{2S - (a + \ell)}{a + \ell}} = \frac{(\ell - a)(a + \ell)}{2S - a - \ell}$.Using the identity $(x - y)(x + y) = x^2 - y^2$,we get $d = \frac{\ell^2 - a^2}{2S - a - \ell}$.Comparing this with the given options,none of the provided expressions match the derived result.

If the first term and the last term of an arithmetic progression are $a$ and $\ell$ respectively,and the sum of all its terms is $S$,then what is its common difference $d$?

Similar Questions

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

If $x, y, z$ are three real numbers with the same sign,then the value of $\frac{x}{y} + \frac{y}{z} + \frac{z}{x}$ lies in which interval?

If the sum of the first four terms of an $A.P.$ is $6$ and the sum of its first six terms is $4$, then the sum of its first twelve terms is

Insert $6$ numbers between $3$ and $24$ such that the resulting sequence is an $A.P.$

If $a, b, c$ are in Arithmetic Progression,then $(a - c)^2 = \dots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module