Mathematical Induction Problem (IB Math HL)

**alireza1992** · June 19th 2012, 08:06 AM

Hi. There was a question in my math book which I couldn't solve. And I would appreciate if anyone could give me the solution.
The question is about mathematical induction and it's from IB math HL textbook.
Bellow is the question (It's in the picture)
$Mathematical Induction Problem (IB Math HL)-math-problem-.png$

**richard1234** · June 19th 2012, 09:57 AM

The base cases n = 1, 2, 3 can be verified to be true. Assume it is true for some n = k. We want to show that

$(k+1) + 2k + 3(k-1) + ... + (k-1)3 + k(2) + (k+1) = \frac{(k+1)(k+2)(k+3)}{6}$

Using our useful "hint," rewrite the LHS as

$k + 2(k-1) + 3(k-2) + ... + (k-1)2 + k + (1+2+...+k+(k+1))$

$=\frac{k(k+1)(k+2)}{6} + (1+2+...+k+(k+1))$

$=\frac{k(k+1)(k+2)}{6} + \frac{(k+1)(k+2)}{2}$

$=\frac{k(k+1)(k+2) + 3(k+1)(k+2)}{6}$

$=\frac{(k+1)(k+2)(k+3)}{6}$

**alireza1992** · June 19th 2012, 10:09 AM

Thanks a a lot

Just one question, how did you get
(k+1)(k+2)/2
from (1+2+...+k+(k+1))
I couldn't follow that part. Could you please explain it a bit?
And thank you for helping me.

**richard1234** · June 19th 2012, 10:13 AM

The sum of the integers from $1$ to $a$ is $\frac{a(a+1)}{2}$ (which can also be proved by induction!)

Therefore the sum of the integers from $1$ to $k+1$ is $\frac{(k+1)(k+2)}{2}$ .

**alireza1992** · June 19th 2012, 10:16 AM

I see now. Thank you so much!

**richard1234** · June 19th 2012, 10:17 AM

No problem

**DEEMATHWHIZ** · January 31st 2013, 11:31 AM

How did you get from k+2(k-1) +3(k-2)..... +(k-1)2+k
(.....
resulting k(k+1)(k=2)/6 ?

Thread: Mathematical Induction Problem (IB Math HL)

LinkBack

Thread Tools

Display

Mathematical Induction Problem (IB Math HL)

Re: Mathematical Induction Problem (IB Math HL)

Re: Mathematical Induction Problem (IB Math HL)

Re: Mathematical Induction Problem (IB Math HL)

Re: Mathematical Induction Problem (IB Math HL)

Re: Mathematical Induction Problem (IB Math HL)

Re: Mathematical Induction Problem (IB Math HL)

Similar Math Help Forum Discussions

Mathematical Induction Problem

Mathematical Induction Problem!

Mathematical Induction Problem

mathematical induction problem

mathematical induction problem no.2

Search Tags