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I need help...

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Hello!

I am asking for help on an AoPS question so I would rather not to be given the answer.

Here is the question:

Compute the sum of: $$101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2$$

Here is what I have:

(a) From the difference of squares, we know that $$(a + b)(a - b) = a^2 - b^2$$. Using this pattern, we can pair up the terms in this sequence and apply:

\begin{align*} 101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2 &= (101 + 97)(101 - 97) + (93 + 89)(93 - 89) + \cdots + (5 + 1)(5 - 1), \\ &= 4(101 + 97) + 4(93 + 89) + ... + 4(5 + 1), \\ &= 4(198 + 182 + 166 + ... + 6). \end{align*}

\begin{align*} 4(198 + 182 + 166 + ... + 6) &= 4(\frac{204 \cdot 12}{2}), \\ &= 4(204 \cdot 6), \\ &= 4(1224), \\ &= 4896. \end{align*}

So, the sum of the sequence $$101^2 - 97^2 + 93^2 - 89^2 + \cdots + 5^2 - 1^2$$, is $$\boxed{4896}$$.

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But I am not sure this is correct, any suggestions?

Jan 6, 2021

2+0 Answers

#1
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This looks accurate to me...

Jan 6, 2021
#2
+23
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Actually, I just figured out what I did wrong, there are 13 terms, not 12... The correct answer is 5304.

Jan 6, 2021