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Prove that 1+1* ""^(1)P(1)+2* ""^(2)P(2)...

Prove that `1+1* ""^(1)P_(1)+2* ""^(2)P_(2)+3* ""^(3)P_(3) + … +n* ""^(n)P_(n)=""^(n+1)P_(n+1).`

Text Solution

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`1+1*(1!)+2*(2!)+3*(3!) +...n*(n!)`
`=1+(2-1)(1!)+(3-1)*(2!)+(4-1)*(3!)+...+{(n+1)-1}*(n!)`
`={1+2*(1!)+3*(2!)+4*(3!)+...+(n+1)*(n!)}-{1!+2!+3!+...+n!}`
`={1+2!+3!+4!+… +n!+(n+1)!}-{1!+2!+3!+...+n!}=(n+1)!.`
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