substitute n=1 and do summation for r and then it is true for n so prove it for n+1 using n
as with most induction problems show that $P(1)=True$ and that $P(n) \Rightarrow P(n+1)$
$P(1) = \left\{\displaystyle{\sum_{r=1}^1}1(1+2r)=1(2\cdot 1 + 1)\right\}=\left\{3=3\right\}=True$
I leave it to you to show that $P(n) \Rightarrow P(n+1)$ i.e. that
$\left\{\displaystyle{\sum_{r=1}^n}n(n+2r)=n(2n+1) \right\} \Rightarrow \left\{\displaystyle{\sum_{r=1}^{n+1}}(n+1)((n+1)+ 2r)=(n+1)(2(n+1)+1)\right\} $
Is easier to prove
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$\displaystyle \sum_{r = 1}^n\{n(n + 2r)\} = n\left\{\sum_{r = 1}^n(n + 2r)\right\} = n\left\{\left(\sum_{r=1}^nn\right) + 2\left(\sum_{r=1}^nr\right)\right\} = n\left(n^2 + 2 * \dfrac{n(n + 1)}{2}\right) =$
$n\left(n^2 + 2 * \dfrac{n^2 + n}{2}\right) = n(n^2 + n^2 + n) = n(2n^2 + n) = n^2(2n + 1) \ne n(2n + 1)\ unless\ n = 0\ or\ n = 1.$
$\displaystyle n = 2 \implies \sum_{r = 1}^2\{2(2 + 2r)\} = 2(2 + 2 * 1) + 2(2 + 2 * 2) = 2 * 4 + 2 * 6 = 8 + 12 = 20 =$
$4 * 5 = 2^2(2 * 2 + 1)\ne 2(2 * 2 + 1).$
$\displaystyle n = 3 \implies \sum_{r = 1}^3\{3(3 + 2r)\} = 3(3 + 2 * 1) + 3(3 + 2 * 2) + 3(3 + 2 * 3) = 3(5 + 7 + 9) = 3 * 21 = 3 * 3 * 7 =$
$3^2(2 * 3 + 1) \ne 3(2 * 3 + 1).$
The student is trying to prove a falsity and so is understandably having trouble. I suggest we ask the student to check what the actual proposition to be proven is.
EDIT: Romsek is correct. RL Brown identified that what was trying to be proved is false several posts back. I am leaving my post up, however, because this student needs it spelled out.
I think this student is struggling enough to warrant an answer.
The intuition behind proofs by induction is this: if something is true for the next integer if it is true for this integer and if that something is also true for 1, then it is true for 2, which means it is true for 3, which means it is true for 4, and so on forever and ever and ever. Got the intuition?
Now for the proof that a proposition P(n) is true for every positive integer n, the structure of a proof by weak mathematical induction is this:
Step 1: Prove P(1) is true.
Now this means that there is at least one positive integer for which P is true. Choose an arbitrary one of such positive integers (call it k). So P(k) is true. This is frequently called the "induction hypothesis." It is not a mere assumption. You have proved in step 1 that such numbers exist.
Step 2: Prove P(k + 1) is true given that P(1) and P(k) are true and k is a positive integer.
Let's do this for your problem.
$\displaystyle Prove\ by\ induction\ that\ n\ is\ any\ positive\ integer \implies \left\{\sum_{r = 1}^nn(n + 2r)\right\} = n^2(2n + 1).$
Now as a practical matter I like to see whether the proposition is true for 2 and 3 before I do the proof so I don't waste time on trying to prove a falsity, but that is NOT part of a formal proof. OK Formal proof
Step 1
$\displaystyle n = 1 \implies \left\{\sum_{r = 1}^nn(n + 2r)\right\} = 1(1 + 2 * 1) = 1(2 * 1 + 1) = 1^2(2 * 1 + 1) = n^2(2n + 1).$
Thus there exists a positive integer k such that $\displaystyle \left\{\sum_{r = 1}^kk(k + 2r)\right\} = k^2(2k + 1) = 2k^3 + k^2.$
Step 2
$\displaystyle \left\{\sum_{r = 1}^{k+1}(k + 1)(\{k + 1\} + 2r)\right\} = \left\{\sum_{r = 1}^k(k + 1)(\{k + 1\} + 2r)\right\} + (k + 1)\{(k + 1) + 2(k + 1)\} =$
$\displaystyle \left\{\sum_{r = 1}^k(k + 1)(\{k + 1\} + 2r)\right\} + 3k^2 + 6k + 3 =$
$\displaystyle \left\{\sum_{r = 1}^kk\{1 + (k + 2r)\right\} + \left\{\sum_{r = 1}^k1(\{k + 1\} + 2r)\right\} + 3k^2 + 6k + 3 =$
$\displaystyle \left\{\sum_{r = 1}^kk\right\} + \left\{\sum_{r = 1}^kk(k + 2r)\right\} + \left\{\sum_{r = 1}^k(k + 1 + 2r)\right\} + 3k^2 + 6k + 3 =$
$\displaystyle k^2 + (2k^3 + k^2) + \left\{\sum_{r = 1}^k(k + 1 + 2r)\right\} + 3k^2 + 6k + 3 =$
$\displaystyle 2k^3 + 5k^2 + 6k + 3 + \left\{\sum_{r = 1}^k(k + 1 + 2r)\right\}=$
$\displaystyle 2k^3 + 5k^2 + 6k + 3 + \left\{\sum_{r = 1}^kk\right\} + \left\{\sum_{r = 1}^k1\right\} + \left\{\sum_{r = 1}^k2r\right\}=$
$\displaystyle 2k^3 + 5k^2 + 6k + 3 + k^2 + k + 2 * \left\{\sum_{r = 1}^kr\right\}=$
$\displaystyle 2k^3 + 6k^2 + 7k + 3 + 2 * \left\{\sum_{r = 1}^kr\right\}=$
$2k^3 + 6k^2 + 7k + 3 + 2 * \dfrac{k(k + 1)}{2} =$
$2k^3 + 6k^2 + 7k + 3 + k^2 + k=$
$2k^3 + 7k^2 + 8k + 3= (k + 1)(2k^2 + 5k + 3) = (k + 1)(k + 1)(2k + 3) = (k + 1)^2(2k + 2 + 1) =$
$(k + 1)^2\{(2(k + 1) + 1\}.$ QED