# ALmost got it, mathematical induction, writing terms seperatley

1. Sep 30, 2006

### mr_coffee

Hello everyone i'm having problems on this last part of mathematical induction. I have to show that the two equations are equal to eachother. The book shows a few examples which i will show below. They are writing the kst term separately from the first k terms.

Heres my problem firstly:
Prove by mathematical induction:
http://img219.imageshack.us/img219/1613/lastscancw1.jpg [Broken]

My Goal is to prove that those 2 equations do infact equal eachother, the boxed equations. Once I write the k term separatley then I go on to substitute from the inductive hypotheiss, then do some algebra.

Examples of writing the terms seperately:
http://img295.imageshack.us/img295/9369/lastscan3cn3.jpg [Broken]

Under my problem i attempted to mimic what they are doing, is that it or no? Any help would be great on explaining what they are doing here!

Last edited by a moderator: May 2, 2017
2. Sep 30, 2006

### nocturnal

it should be

$$\sum_{i = 1}^{k + 2} i2^i = \left(\sum_{i = 1}^{k+1} i2^i \right) + (k+2)2^{k+2}$$

As a simple example consider:

$$\sum_{i=1}^{4}i = 1 + 2 + 3 + 4 = (1 + 2 + 3) + 4 = \left(\sum_{i=1}^{3}i \right) + 4$$

more generally, if f is a function defined on the integers, and a, b, and c are integers with $a \leq b \leq c$ then,

\begin{align*} \sum_{i = a}^{c} f(i) &= f(a) + f(a+1) + \ldots + f(b-1) + f(b) + f(b+1) + \ldots + f(c-1) + f(c) \\ &= \left( f(a) + f(a+1) + \ldots + f(b-1) \right) + \left( f(b) + f(b+1) + \ldots + f(c-1) + f(c) \right) \\ &= \left( \sum_{i = a}^{b-1} f(i) \right) + \left( \sum_{i = b}^{c} f(i) \right) \end{align*}

in regards to your problem, f is the function
$$f(k) = k2^k$$
and a = 1, b = c = k + 2

Your proof up to this point looks great by the way, very nicely written.

Last edited: Sep 30, 2006
3. Oct 2, 2006

### mr_coffee

Thanks nocturnal, that explanation was very helpful. Sorry about the delayed responce.

I'm now having the following troubles, i can't seem to manipulate the left hand side of the equation to look like the right hand side.

This is my work:

http://img187.imageshack.us/img187/1449/lastscanfj5.jpg [Broken]

I'm suppose to get that to look like right hand side of the equation, (k+1)*2^(k+3) + 2, any help or suggestions would be great! I tried to factor and expand, and they dont' seem to work or maybe i can't see somthing.

THanks!

Last edited by a moderator: May 2, 2017