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hartnn
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@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 :)
Bugay♥
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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)
Bugay♥
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thanks @hartnn ..
DLS
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Satisfy by k
Satisfy by k+1
hartnn
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welcome :)
do you know general steps for proving an identity by mathematical induction ?
Bugay♥
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basis of induction , induction hypothesis and proof of induction..
hartnn
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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 .
Bugay♥
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3 is divisible by 3
then?
Bugay♥
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??
Bugay♥
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hartnn : i thought you will help me.. ???
hartnn
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i am sorry, i keep on getting disconnected..
Bugay♥
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oh its ok..
hartnn
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well, next step is to assume the result true for n=k
so, k^3+2k is divisible by 3---->(A)
hartnn
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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
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using the fact that k^3+2k is divisible by 3
can you do that ? try it...
Bugay♥
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no i cant :(( can you do it for me?
Bugay♥
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@Tushara : hello..
Bugay♥
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@hartnn : its okey thank you so much..
Tushara
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hey m doing the problem... ill help u out in a bit
Bugay♥
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@Tushara : i wish you can help me with this..
Tushara
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Tushara
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does the second proof have any rule on n? like n>1?
Tushara
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the second proof is not true for n=1
Bugay♥
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no..
Tushara
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well then u cant prove the second one.... its just not true
Bugay♥
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are you sure??
Bugay♥
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let me check the given..
Tushara
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yeah m sure
Bugay♥
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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
Tushara
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2^n=n^2(2n^2-1) for n=1 which is not true
Tushara
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its not true for n=2 either
Kira_Yamato
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Bugay♥
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oops im sorry the given was wrong.. it should be 2+2^2+2^3+ . . . + 2^n = 2^(n+1) - 2
Tushara
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okay,... well its a very easy proof... prove true for n=1, assume true for n=k, then prove true for k+1
Bugay♥
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it is now true for n=1 right?? then? what i am going to do?
Tushara
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assume true for n=k
Bugay♥
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@Kira_Yamato : still i thank you..
Tushara
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now prove true for n=k+1
Bugay♥
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then?? i find difficulty in proof of induction :((
Tushara
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have u practiced any induction problems before? if u have some induction examples in ur math text book... please go thru them
Tushara
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all u have to do is this:
prove that
2^(n+1)-2+2^(n+1)=2(n+2)-2
Bugay♥
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my teacher dont taught mathematical induction to us.. i havent encounter it before..
Tushara
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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
Tushara
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2^(n+1)-2+2^(n+1)=2^(n+2)-2
sorry i typed it up wrong before
Bugay♥
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what should i prove? if it is equal?
Tushara
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yes its equal... dats all u have to do for that question
Bugay♥
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oh okey.. thanks..