The value of (2^1P_0 - 3^2P_1 + 4^3P_2 - up to 51^th term) + (1!-2!+3!-up to 51^th term) i…

Question

The value of $\left(2\cdot{}^1P_0 - 3\cdot{}^2P_1 + 4\cdot{}^3P_2 - \cdots	ext{up to }51^{	ext{th}}	ext{ term}ight) + \left(1!-2!+3!-\cdots	ext{up to }51^{	ext{th}}	ext{ term}ight)$ is equal to:

Accepted Answer

Note $(r+1)\cdot{}^rP_{r-1} = (r+1)\cdot\dfrac{r!}{1!}=(r+1)!$

So the first sum $= 2!-3!+4!-\cdots+52!$ (51 terms), and adding the second sum $1!-2!+3!-\cdots+51!$ (also telescoping):

$$[2!-3!+4!-\cdots+52!] + [1!-2!+3!-\cdots+51!] = 52!+1! = 52!+1$$

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