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Logarithmic Differentiation

If 1 + 2x + 3...

Question

If ${(1 + 2 x + 3 x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}$ , then $a_{1}$ equals

A

$10$

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B

$20$

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C

$210$

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D

None of these

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Solution

The correct option is B
$20$

Explanation for the correct option:
Find the value of $a_{1}$ :
Given, ${(1 + 2 x + 3 x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}$
Differentiate it with respect to $x$ , we get
$10 {(1 + 2 x + 3 x^{2})}^{9} (2 + 6 x) = a_{1} + 2 a_{2} x + . . . + 20 a_{20} x^{19}$
Put $x = 0$
$10 (2) = a_{1}$
$∴ a_{1} = 20$
Hence, Option ‘B’ is Correct.

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