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Formula for Sum of n Terms of an AP

In an Arithme...

Question

In an Arithmetic sequence, $S_{n}$ represents the sum to $n$ terms, what is $S_{n} - S_{n - 1}$ ?

t_{1} + t_{2} + . . . . t_{n - 1}

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S_{n - 1}

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n - 2 \sum n = 1 t_{n}

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t_{n}

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Solution

The correct option is D $t_{n}$
We know,
$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
$∴ S_{n - 1} = \frac{n - 1}{2} [2 a + (n - 2) d]$

$∴ S_{n} - S_{n - 1} = \frac{n}{2} [2 a + (n - 1) d] - \frac{n - 1}{2} [2 a + (n - 2) d]$

$= \frac{1}{2} [2 a n + n^{2} d - n d - 2 a n + 2 a - n^{2} d + 3 n d - 2 d]$

$= \frac{1}{2} [2 a + 2 n d - 2 d]$

$= a + (n - 1) d]$

$= t_{n}$
$(A n s \to D)$

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