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Arithmetic Progression

The sum of th...

Question

The sum of the first nineteen terms of an A.P. $a_{1}, a_{2}, a_{3}$ .......... if it is known that $a_{4} + a_{8} + a_{12} + a_{16} = 224$ is .......... .

1064

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896

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532

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448

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Solution

The correct option is A $1064$
We know that $n^{t h}$ term of an AP is given as, $a_{n} = a + (n - 1) d$

And sum of first $n$ terms, $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ , where $a$ and $d$ are the first term amd common difference of an AP.
Since, $a_{4} + a_{8} + a_{12} + a_{16} = 224$

$∴ a + 3 d + a + 7 d + a + 11 d + a + 15 d = 224 \Rightarrow 4 a + 36 d = 224 \Rightarrow a + 9 d = 56$
Now
$S_{19} = \frac{19}{2} [2 a + (19 - 1) d] = 19 [a + 9 d] = 19 [56] = 1064$

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