I am trying to prove $2+4+6+\cdots +2n=n^2+n$ by mathematical induction. I followed all the steps and the $P_{k+1}$ was $2+4+6+\cdots+2(k+1)=(k+1)^2+k+1$ Starting from the left hand side of the equation I have solved till $k^2+k+2(k+1)$. Now I am stuck here. I don't know how to do it further. Please guide me thanks.
$2+4+6+\cdots+2n=n^2+n$ by mathematical induction
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induction
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3The formulas are wrong, $2^n$ and $2^{k+1}$ should read $2n$ and $2(k+1)$. – 2011-11-06
2 Answers
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It looks as if you need $2+4+6+\cdots+2n$ rather than $2+4+6+\cdots+2^n$.
Look at $ \underbrace{2+4+6+\cdots+2k}+2(k+1). $ The induction hypothesis tells you what to do with the part over the underbrace. Then massage it a bit with some simple algebra.
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0@Gilles: Thanks for pointing it. Its a wrong proposition – 2011-11-07
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$k^2+k+2(k+1)=k^2+3k+2=k^2+2k+1+(k+1)= (k+1)^2+(k+1)$