Compute $1^2 + 3^2+ 5^2 + \cdots + (2n-1)^2$ by mathematical induction

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I am doing mathematical induction. I am stuck with the question below. The left hand side is not getting equal to the right hand side.
Please guide me how to do it further.

$1^2 + 3^2+ 5^2 + \cdots + (2n-1)^2 = \frac{1}{3}n(2n-1)(2n+1)$.

Sol:

$P(n):\ 1^2 + 3^2 + 5^2 + \cdots + (2n-1)^2 = \frac{1}{3}n(2n-1)(2n+1)$.

For $n=n_1 = 1$

$$P(1) = \frac{1}{3}{3} = (1)^2.$$ Hence it is true for $n=n_0 = 1$.

Let it be true for $n=k$ $$P(k): 1^2 + 3^2 + 5^2 + \cdots + (2k-1)^2 = \frac{1}{3}k(2k-1)(2k+1).$$ We have to prove that it is true for $P(k+1)$. $$P(k+1) = 1^1+3^2+5^2+\cdots+(2k+1)^2 = \frac{1}{3}(k+1)(2k+1)(2k+3)\tag{A}.$$

Taking LHS: $$\begin{align*} 1^2 + 3^2 + 5^2 + \cdots + (2k+1)^2 &= 1^2+3^2+5^2+\cdots + (2k+1)^2\\ &= 1^2 + 3^2 + 5^2 + \cdots + (2k-1)^2 + (2k+1)^2\\ &= \frac{1}{3}k(2k-1)(2k+1) + (2k+1)^2\\ &=\frac{k(2k-1)(2k+1)+3(2k+1)^2}{3}\\ &=\frac{(2k+1)}{3}\left[k(2k-1) + 3(2k+1)\right]\\ &=\frac{(2k+1)}{3}\left[2k^2 - k + 6k + 3\right]\\ &=\frac{1}{3}(2k+1)(2k^2 +5k + 3)\\ &=\frac{1}{3}(2k+1)(k+1)\left(k+\frac{3}{2}\right) \tag{B} \end{align*}$$

EDIT:

Solving EQ (A):

$=(1/3)(2k^2+5K+3) (2K+1) \tag{C}$

Comparing EQ(B) and EQ(C)

Hence proved that it is true for $n = k+1.$

Thus the proposition is true for all $n >= 1$.

Thanks.

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2 Answers

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NOTE: I am not saying anything different, before someone start commenting that my answer is not any different.

$$ \begin{align*} 1^2 + 3^2 + 5^2 + \cdots + (2k+1)^2 &= 1^2+3^2+5^2+\cdots + (2k+1)^2\\ &= 1^2 + 3^2 + 5^2 + \cdots + (2k-1)^2 + (2k+1)^2\\ &= \frac{1}{3}k(2k-1)(2k+1) + (2k+1)^2\\ &=\frac{k(2k-1)(2k+1)+3(2k+1)^2}{3}\\ &=\frac{(2k+1)}{3}\left[k(2k-1) + 3(2k+1)\right]\\ &=\frac{(2k+1)}{3}\left[2k^2 - k + 6k + 3\right]\\ &=\frac{1}{3}(2k+1)(2k^2 +5k + 3)\\ &=\frac{1}{3}(2k+1) \hspace{3pt}\left[(k+1)(2k+3)\right] \\ &= \frac{1}{3} (k+1)(2(k+1)-1)(2(k+1)+1) \end{align*} $$

The last line shows that the result is true for $n=k+1$

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Everything is OK except for the very last line. You somehow lost a factor of two. The penultimate line is already the result you want, since $2k^2+5k+3=(k+1)(2k+3)$.

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