First slide

Binomial theorem for positive integral Index

Question

If $(1 + x) (1 + x + x^{2}) (1 + x + x^{2} + x^{3}) \dots (1 + x$ $+ x^{2} + \dots + x^{n}) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{m} x^{m}$ then the value of $a_{1}$ is

Moderate

A

$m + 1$
B

$(n + 1)$
C

$n$
D

$m$

Solution

$a_{1} =$ coefficient of $x$ in
$(1 + x) (1 + x + x^{2}) (1 + x + x^{2} + x^{3}) \dots (1 + x + x^{2} + \dots + x^{n})$
$\begin{array}{l} = coefficient of x in (1 + x)^{n} \\ =^{n} C_{1} = n \end{array}$

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