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- anonymous

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- hartnn

@Bugay♥ Hi :)
\(\huge \color{red}{\text{Welcome to Open Study}}\ddot\smile\)
Post a specific question, and we'll try our best to help you :)

- anonymous

by the method of mathematical induction prove that the following are valid for all positive values of n.
1.) n^3+2n is divisible by 3
2.) 2+2^2+2^3+ . . . + 2^n = n^2(2n^2-1)

- anonymous

thanks @hartnn ..

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## More answers

- DLS

Satisfy by k
Satisfy by k+1

- hartnn

welcome :)
do you know general steps for proving an identity by mathematical induction ?

- anonymous

basis of induction , induction hypothesis and proof of induction..

- hartnn

First we prove the result for n= 1
so, put n=1 in n^3+2n and check whether the answer is divisible by 3 .

- anonymous

3 is divisible by 3
then?

- anonymous

??

- anonymous

hartnn : i thought you will help me.. ???

- hartnn

i am sorry, i keep on getting disconnected..

- anonymous

oh its ok..

- hartnn

well, next step is to assume the result true for n=k
so, k^3+2k is divisible by 3---->(A)

- hartnn

now, using (A), we need to prove the result for n=k+1
that is, prove (k+1)^3+2(k+1) is divisible by 3

- hartnn

using the fact that k^3+2k is divisible by 3
can you do that ? try it...

- anonymous

no i cant :(( can you do it for me?

- anonymous

@Tushara : hello..

- anonymous

@hartnn : its okey thank you so much..

- anonymous

hey m doing the problem... ill help u out in a bit

- anonymous

@Tushara : i wish you can help me with this..

- anonymous

##### 1 Attachment

- anonymous

does the second proof have any rule on n? like n>1?

- anonymous

the second proof is not true for n=1

- anonymous

no..

- anonymous

well then u cant prove the second one.... its just not true

- anonymous

are you sure??

- anonymous

let me check the given..

- anonymous

yeah m sure

- anonymous

we have to put n=1 to n^2(2n^2-1) right?? and if it is equal to 1 .. the theorem is true for n=1

- anonymous

2^n=n^2(2n^2-1) for n=1 which is not true

- anonymous

its not true for n=2 either

- Kira_Yamato

##### 1 Attachment

- anonymous

oops im sorry the given was wrong.. it should be 2+2^2+2^3+ . . . + 2^n = 2^(n+1) - 2

- anonymous

okay,... well its a very easy proof... prove true for n=1, assume true for n=k, then prove true for k+1

- anonymous

it is now true for n=1 right?? then? what i am going to do?

- anonymous

assume true for n=k

- anonymous

@Kira_Yamato : still i thank you..

- anonymous

now prove true for n=k+1

- anonymous

then?? i find difficulty in proof of induction :((

- anonymous

have u practiced any induction problems before? if u have some induction examples in ur math text book... please go thru them

- anonymous

all u have to do is this:
prove that
2^(n+1)-2+2^(n+1)=2(n+2)-2

- anonymous

my teacher dont taught mathematical induction to us.. i havent encounter it before..

- anonymous

if u cant prove the above equation^ den its best for u to not study ahead and wait for ur teacher to teach u... just see if u can prove the above

- anonymous

2^(n+1)-2+2^(n+1)=2^(n+2)-2
sorry i typed it up wrong before

- anonymous

what should i prove? if it is equal?

- anonymous

yes its equal... dats all u have to do for that question

- anonymous

oh okey.. thanks..

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