If P = 2 2+ 6 2+ 10 2+ 14 2+ _______ + 94 2and Q = 1 2+ 5 2+ 9 2+ _______ + 81 2, then what is the value of P – Q?
26075
Step 1 — Term counts: P has bases 2, 6, ..., 94 i.e. \(4n-2\) for \(n=1..24\). Q has bases 1, 5, ..., 81 i.e. \(4n-3\) for \(n=1..21\).
Step 2 — Pair first 21 terms:
\[(4n-2)^2-(4n-3)^2 = 8n-5\]
\[\sum_{n=1}^{21}(8n-5) = \tfrac{21}{2}(3+163) = 1743\]
Step 3 — Add leftover terms of P: \(86^2+90^2+94^2 = 7396+8100+8836 = 24332\).
Therefore P − Q = 1743 + 24332 = 26075.
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