Error in finding sum of $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$

Question

To find sum of the series $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$
My approach,
Let S=$1^2+2^2+3^2 + \cdots +n^2$
If $n$ is even
S=$(1-2)^2+(3-4)^2+ \cdots +[(n-1)-n]^2+2(1 \cdot 2+3 \cdot 4+5 \cdot 6+ \cdots \text{to}\space \dfrac{n}{2}\space \text{terms})$
=$\frac{n}2+2(1 \cdot 2+3 \cdot 4+5 \cdot 6+ \cdots \text{to}\space \dfrac{n}{2}\space \text{terms})$
And we know,
$$S=\frac{n(n+1)(2n+1)}6$$
Therefore,$$2(1 \cdot 2+3 \cdot 4+5 \cdot 6+ \cdots \text{to}\space \frac{n}2\space \text{terms}) =\frac{n(n+1)(2n+1)}6-\frac{n}2$$
or $$2(1 \cdot 2+3 \cdot 4+5 \cdot 6+ \cdots \text{to}\space \frac{n}2\space \text{terms})=\frac{n}2.\frac{2n^2+3n-2}3$$
or $$(1 \cdot 2+3 \cdot 4+5 \cdot 6+ \cdots \text{to}\space \frac{n}2\space \text{terms})=\frac{n(n+2)(2n-1)}{24}$$
But, this is the sum of $\dfrac{n}{2}$ terms. To get sum of $n$ terms, we replace $n$ by $2n$
Thus,
$$(1 \cdot 2+3 \cdot 4+5 \cdot 6+ \cdots \text{to}\space n\space \text{terms})=\frac{n(n+1)(4n-1)}6$$
So, if input of $n$ is even result should be correct,
if $n=2$, sum (from formula) = $7$
However, actual sum is $14$ (=$1 \cdot 2+3 \cdot 4$).
What is the error in the above approach?
EDIT: I rechecked my calculations and also have written the steps but still the answer isn't coming right. Please tell in which step the problem lies.
Thanks for your time and patience.

Answer 1

Note that $$\frac{n(n + 1)(2n + 1)}{6} - \frac{n}{2} = \frac{n}{2}\frac{2n^{2} + 3n - 2}{3}$$ and if you divide this expression by $2$ you are supposed to get $$\frac{n(n + 2)(2n - 1)}{12}$$ but you get mistakenly $$\frac{n(n + 2)(2n - 1)}{24}$$ This is the source of error. If you fix this problem you will get the right formula $$(1\cdot 2) + (2\cdot 4) + \cdots + \{(2n - 1)\cdot (2n)\} = \frac{n(n + 1)(4n - 1)}{3}$$

Answer 2

$$\sum_{m=1}^n (2m-1)(2m) = \frac{1}{3}n(n+1)(4n-1)$$

Answer 3

Interesting approach and it will work for all values of n if you do it right.

The sum of first n terms in the sequence 1.2 + 3.4 + ... is calculated by the taking first sum of 2n terms of the sum of squares sequence (S in your case), subtracting n from it and taking the half.

say n = 2 sum of first 4 squares is 30 subtract 2 and take half 14.

for n = 3 sum of first 6 squares is 91 subtract 2 and take half 44.

and so on.

Your method works, however when you write "Combining the above results, we get," there you have done a calculation mistake.

There are simpler ways of doing the problem, however I like your approach.