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Question

Find the value of :
1×1!+2×2!+3×3!+.......+n×n!

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Solution

1×1!=(21)×1!=2×1!1×1!=2!1!
2×2!=(31)×2!=3×2!2!=3!2!
3×3!=(41)×3!=4×3!3!=4!3!
n×n!=(n+11)×n!=(n+1)(n!)n!=(n+1)!n!
Summing up all these terms, we get
=2!1!+3!2!+4!3!+...+(n+1)!n!
=(n+1)!1!


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