$1^2 - 2^2 + 3^2 - 4^2 + \dots + 1999^2$ using closed form formula for sum of squares of first $n$ natural numbers

Question

The question is simple, the solution, not so much

Q.Find the sum of the given expression $1^2- 2^2 + 3^2 - 4^2 + \dots + 1999^2$

My idea is we know $1^2 + 2^2 + 3^2 + 4^2 + \dots + n^2 = \frac{n(n + 1)(2n + 1)}{6}$

So for $n=1999$ I get the sum as $2,66,46,67,000$

From this I need to subtract the squares of the even terms twice because subtracting once leaves with only the sum of the squares of the odd nos.

I observed something :

$2^2 + 4^2 + 6^2 + \dots + 1998^2 = (2 \cdot 1)^2 + (2 \cdot 2)^2 + \dots + (2 \cdot 999)^2$

Therefore to obtain the sum of the square of the even terms, I can take $4$ as common and use the aforementioned formula for $n=999$ and multiply it by $4$.

therefore sum of square of even terms = $1,33,13,34,000$

I need to subtract this sum twice to get the answer, because subtracting once simply leaves me with the sum of the squares of the odd numbers.

The answer is now $1999000$, which still doesn't match the answer key. Can someone explain where I am going wrong ?

Answer 1

If you first add all the squares: $$ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + \cdots $$ and then subtract the even squares: $$ \phantom{1^2} - 2^2\phantom{ + 3^2} - 4^2 \phantom{+ 5^2} - 6^2 \phantom{-}\cdots $$ then you are left with only the odd squares: $$ 1^2 \phantom{+ 2^2} + 3^2 \phantom{+ 4^2} + 5^2 \phantom{+ 6^2} + \cdots $$ You need to subtract the even squares one more time to get $$ 1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \cdots $$

Answer 2

Use the decomposition of squares into triangular numbers, $n^2=T_{n-1}+T_n$, and telescope: $$1^2-2^2+3^2-\cdots+1999^2=1-(1+3)+(3+6)-(6+10)+\cdots+(T_{1998}+T_{1999})$$ $$=T_{1999}=\frac{1999×2000}2=1999000$$

Answer 3

There is a slight mistake

From this I need to subtract the sum of squares of even terms.

No. You have to subtract twice of above because

$1^2-2^2+3^2+\cdots+1999^2=(1^2+2^2+3^2+\cdots+1999^2)-2(2^2+4^2+6^2+1998^2)$

Answer 4

Looking at your work, if you only subtract the sum of the even terms from the sum of all terms, you are only left with the sum of the odd terms, not the alternating sum you have shown above.

Hint: How many times do I need to subtract a term from a summation to get it to go from + to - ?

Answer 5

We can use the identity $$(x+1)^2-x^2=x+1+x$$ to tackle the sum in 2 simpler ways:

Way 1 $$ \begin{aligned}&1^2 - 2^2 + 3^2 - 4^2 + 5^2-... -1998^2+ 1999^2 \\=& 1+(2+3)+(4+5)+…+(1998+1999) \\= &19990000\end{aligned}$$

Way 2 $$ \begin{aligned}&(1^2 - 2^2 + 3^2 - 4^2 + 5^2-6^2...-1998^2)+ 1999^2 \\=&-((1+2)+(3+4)+(5+6)+…+(1997+1998))+1999^2 \\= &19990000\end{aligned}$$