Calculate the sum of the series $1 + 3 + 5 + 7 + . . . .2 n - 1$ .

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Solution

$1 + 3 + 5 + 7 + . . . .2 n - 1$
The given series is in AP
$a = 1, d = 2$
The sum of series is

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

$= \frac{n}{2} [2 + (n - 1) 2]$

$= \frac{n}{2} (2 n) = n^{2}$

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