Can anybody simplify it? Show me the way of simplification.
The expression is as follows:
$$F(x) = 1 *(1!+x)+2*(2!+x)+ ..+x*(x!+x)$$
for a positive integer $x$ I've tried but nothing got.
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$r$th term $=r(r!+x)=r\cdot r!+rx=(r+1-1)r!+rx=\underbrace{(r+1)!-r!}+rx$
The terms underbraced is Telescoping and $\sum_{r=1}^xrx=x\sum_{r=1}^xr=x\dfrac{x(x+1)}2$
lab bhattacharjee
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awesome answer (+1) – Shobhit Feb 08 '15 at 16:04
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Thankyou boss for quick response and in so much easy way – Frustrated Feb 08 '15 at 16:11
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@Frustrated, See http://en.wikipedia.org/wiki/Telescoping_series – lab bhattacharjee Feb 08 '15 at 16:14
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@labbhattacharjee, Can you tell me , value of F(4) for better understanding of "Telescoping" in this answer. – Frustrated Feb 08 '15 at 16:19
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@Frustrated, If you have understood my method, you should be able to find $F(x)=?$ then set $x=4$ – lab bhattacharjee Feb 08 '15 at 16:34
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@labbhattacharjee, F(4)=40, is it? or F(4)=159 – Frustrated Feb 08 '15 at 16:36