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

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(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}{

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None of these

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(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 $d$?

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