Question : If $P=2^2+6^2+10^2+14^2+....+ 94^2$ and $Q=1^2+5^2+9^2+...+81^2$, then what is the value of $P - Q$?
Option 1: 24645
Option 2: 26075
Option 3: 29317
Option 4: 31515
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Correct Answer: 26075
Solution :
Given: $P=2^2+6^2+10^2+14^2+....+94^2$ and $Q=1^2+5^2+9^2+...+81^2$
$⇒P-Q = (2^{2}-1^{2})+(6^{2}-5^{2})+(10^{2}-9^{2})+....+(82^{2}-81^{2})+86^{2}+90^{2}+94^{2}$
$⇒P-Q=3+11+19+27+.......+163(\text{A.P. with 21 terms})+86^{2}+90^{2}+94^{2}$
$⇒P-Q=\frac{21}{2}[2×3+(21–1)8]+86^{2}+90^{2}+94^{2}$
$⇒P-Q=\frac{21}{2}[6+(20)8]+86^{2}+90^{2}+94^{2}$
$⇒P-Q=\frac{21}{2}[166]+(90-4)^{2}+90^{2}+(90+4)^{2}$
$⇒P-Q=1743+(90^{2}-2×90×4+4^{2})+90^{2}+(90^{2}+4^{2}+2×90×4)$
$⇒P-Q=1743+8100+16+8100+8100+16$
$\therefore P-Q=26075$
Hence, the correct answer is 26075.
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