Proof by Induction
Prove that:
n∑1
(r2 + 1)r! = n(n+ 1)!
Case for n = 1:
2× 1 = 1× 2
It works! Which is nice.Assuming this beast is true when n = k:
k∑1
(r2 + 1)r! = k(k + 1)!
Now working out what’s going on when n = k + 1:(Note that we know, by the definition of what sums are, that,
p+1∑1
f(r) =
p∑1
f(r) + f(p+ 1)
(Also note that we know that, by the definition of what a factorial is:
(q + 1)! = (q + 1)(q!)
∴k+1∑1
(r2 + 1)r! =k∑1
(r2 + 1)r! + ((k + 1)2 + 1)(k + 1)!
= k(k + 1)! + (k2 + 2k + 2)(k + 1)!
= (k2 + 3k + 2)(k + 1)!
= (k + 2)(k + 1)(k + 1)!
= (k + 1)(k + 2)!
∴ by induction,∑n
1 (r2 + 1)r! = n(n+ 1)!.
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