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If (1+x+x^2++x^p)^n=a0+a1x+a2x^2++a(n p)...

If `(1+x+x^2++x^p)^n=a_0+a_1x+a_2x^2++a_(n p)x^(n p),` then find the value of `a_1+2a_2+3a_3+ddot+n pa_(n p)dot`

Text Solution

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`(1+x+x^(2)+"….."+x^(p))^(n)=a_(0) + a_(1)x+"…."+a_(np)x^(np)`
Differentiating both side w.r.t. , we get
`n(1+x+x^(2)+"……"+x^(p))^(n-1)(1+2x+"….."+px^(p-1))`
`= a_(1)+2a_(2)x+"….."+npa_(np)x^(np-1)`
Now put `x = 1`
`:. a_(1) + 2a_(2) + "......." np a_(np) = n(p+1)^(n-1)(1+2+"...."+p)`
`= (n(p+1)^(n).p)/(2)`
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