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Algebra of Limits

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Question

$300 \sum r = 0 a_{r} x^{r} = (1 + x + x^{2} + x^{3})^{100}$ . If $a = 300 \sum r = 0 a_{r}$ , then $300 \sum r = 0 r a_{r}$ , is equal to

A

300 a

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B

100 a

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C

150 a

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D

75 a

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Solution

The correct option is D $150 a$
Substituting $x = 1$ is $(1 + x + x^{2} + x^{3})^{100}$ , we get
$a = 4^{100}$
Differentiating $(1 + x + x^{2} + x^{3})^{100}$ , with respect to $x$ , we get
$100 (1 + x + x^{2} + x^{3})^{99} (1 + 2 x + 3 x^{2})$
$= \sum r a_{r}$
Substituting $x = 1$ , in the above equation, we get
$\sum r a_{r} = 100 (4^{99}) (6)$
$= 600 (4^{99})$
$= \frac{600 (4^{100})}{4}$
$= 150 (4^{99})$
$= 150 (a)$

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