If (1+x)1+x+x21+x+x2+x3…1+x+x2+…+xn=a0+a1x+a2x2+…+amxmthen value of a1 is

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

$a 1$ = coefficient of x in

$(1 + x) (1 + x + x^{2}) (1 + x + x^{2} + x^{3}) \dots (1 + x + \dots + x^{n})$

= coefficient of x in $\underset{n times}{\underset{⏟}{(1 + x) (1 + x) \dots (1 + x)}}$

= coefficient of x in ${(1 + x)}^{n}$

$= {}^{n}C_{1} = n$

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