+23 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?
