For a series S = 1 - 2 + 3 - 4 + dots up to n | vedclass

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(D) The series is $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms.Case $1$: If $n$ is even,let $n = 2m$.$S = (1 - 2) + (3 - 4) + \dots + ((2m - 1) - 2m)$$

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Both statements are true,and statement $-1$ is the correct explanation of statement $-2$.

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(D) The series is $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms.Case $1$: If $n$ is even,let $n = 2m$.$S = (1 - 2) + (3 - 4) + \dots + ((2m - 1) - 2m)$$S = (-1) + (-1) + \dots + (-1)$ ($m$ times)$S = -m = -\frac{n}{2}$.Case $2$: If $n$ is odd,let $n = 2m + 1$.$S = (1 - 2) + (3 - 4) + \dots + ((2m - 1) - 2m) + (2m + 1)$$S = -m + (2m + 1) = m + 1$.Since $n = 2m + 1$,we have $m = \frac{n - 1}{2}$.$S = \frac{n - 1}{2} + 1 = \frac{n + 1}{2}$.Since the sum depends on whether $n$ is even or odd (Statement $-1$ is true) and the formula for even $n$ is $-\frac{n}{2}$ (Statement $-2$ is true),and the dependence on parity is the reason for the different formulas,Statement $-1$ is the correct explanation for Statement $-2$.

For a series $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms,
Statement $-1$: The sum of the series is always dependent on the value of $n$,i.e.,whether it is even or odd.
Statement $-2$: The sum of the series is $-\frac{n}{2}$ when $n$ is any even integer.

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For a series $S = 1 - 2 + 3 - 4 + \dots$ up to $n$ terms,Statement $-1$: The sum of the series is always dependent on the value of $n$,i.e.,whether it is even or odd.Statement $-2$: The sum of the series is $-\frac{n}{2}$ when $n$ is any even integer.