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The coefficient of x in the expansion of (1+x)(1+2x)(1+3x)....

Question

The coefficient of $x$ in the expansion of $(1 + x) (1 + 2 x) (1 + 3 x) .... (1 + 100 x)$ is

5050

10100

5151

4950

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The coefficient of

x

in the expansion of

(1 + x) (1 + 2 x) (1 + 3 x) .... (1 + 100 x)

11 mins ago

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Text solutionVerified

(1 + x) (1 + 2 x) (1 + 3 x) ...... (1 + 100 x)

We are required to find the coefficient of

^{'} x^{'}

in the above expansion.

Now, first multiply the first two terms

(1 + x) (1 + 2 x) = 1 + x + 2 x + 2 x^{2} = 1 + (1 + 2) x + 2 x^{2}

Similarly multiply the first three terms

(1 + x) (1 + 2 x) (1 + 3 x) = 1 + (1 + 2 + 3) x + 11 x^{2} + 6 x^{3}

So in the expansion of

(1 + x) (1 + 2 x) (1 + 3 x) ...... (1 + 100 x)

coefficient of

x

should be

(1 + 2 + 3 + ........ + 100)

We know that the sum of the first

^{'} n^{'}

natural numbers is given by the expression

\frac{n ( n + 1 )}{2}

∴

Coefficient of

x

in the above expansion is

= \frac{100 \times 101}{2} = 5050

Correct answer : Option A.

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Question Text	The coefficient of $x$ in the expansion of $(1 + x) (1 + 2 x) (1 + 3 x) .... (1 + 100 x)$ is
Answer Type	Text solution:1
Upvotes	150