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Mathematical Induction Problem 2 (IB Math HL)

Hello.

I have another question about mathematical induction. I have problem dealing with factorial notation in mathematical induction problems. Here are two question which I couldn't solve. I'd be grateful if you could help me to figure out how to solve them.

Attachment 24139

Re: Mathematical Induction Problem 2 (IB Math HL)

Re: Mathematical Induction Problem 2 (IB Math HL)

Re: Mathematical Induction Problem 2 (IB Math HL)

Re: Mathematical Induction Problem 2 (IB Math HL)

Proof to part b:

Base: Where n=1:

LHS=1/2!

RHS=((1+1)!-1)/(1+1)!=(2!-1)/2!=1/2!=LHS

Now we assume the statement holds true for n=k, and show that it holds true for n=k+1

So we have

1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!=((k+1)!-1)/(k+1)!

And want to show that

1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!+((k+1))/(k+2)!=((k+2)!-1)/(k+2)!

LHS=1/2!+2/3!+3/4!+4/5!+⋯+k/(k+1)!+((k+1))/(k+2)!

=((k+1)!-1)/(k+1)!+((k+1))/(k+2)!

=((k+2)!(k+1)!-(k+2)!+(k+1)(k+1)!)/(k+1)!(k+2)!

=((k+2)!(k+1)!-(k+2)(k+1)!+(k+1)(k+1)!)/(k+1)!(k+2)!

=((k+2)!-(k+2)+(k+1))/(k+2)!

=((k+2)!-k+k-2+1)/(k+2)!

=((k+2)!-1)/(k+2)!

Q.E.D.