Show, by way of an appropriate substitution(s), that if p = p! / (p-1)! is true,

Question

Show, by way of an appropriate substitution(s), that if p = p! / (p-1)! is true, then it is true that p + 1 = (p + 1)! / p!. That is, start with p + 1 and try to end up with (p + 1)! / p! by using a variety of substitutions you know to be true (i.e. forms of p = p! / (p-1)!). Valid substitutions could include p = p! / (p-1)!; (p - 1)! = p! / p; p! = (p - 1)!; etc. For instance, one way to begin would be to write the following: p + 1 = p! / (p - 1)! + 1 = p! + (p - 1)! / (p - 1)! = p! + p! / p / p! / p = p middot p! + p! / p / p! / p = p!(p + 1) / p!, etc.

Explanation / Answer

(p+1)! = (p+1) (p) (p-1) (p-2) ... (2)(1) = (p+1)(p!) therefore (p+1)!/p! = p+1.

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