1. ## A power question....

12 means I x 1, 22 means 2 x 2, 32means 3 x 3, and so forth. 12+ 22+ 32+ 42 + ... + 252= 5525, and 22+ 42+ 62 + 82 + ... + 502 = N. Find the value of N.

Now we have been told the answer is 22,100; but I don't know how to get there the fast way, apart from actually doing the equation fo each even number until I get to 50. And then adding them all up!

Would anyone be able to explain to a me? Thank you.

2. Originally Posted by ronaldj
12 means I x 1, 22 means 2 x 2, 32means 3 x 3, and so forth. 12+ 22+ 32+ 42 + ... + 252= 5525, and 22+ 42+ 62 + 82 + ... + 502 = N. Find the value of N.

Now we have been told the answer is 22,100; but I don't know how to get there the fast way, apart from actually doing the equation fo each even number until I get to 50. And then adding them all up!

Would anyone be able to explain to a me? Thank you.

Hint:- use
$\boxed { 1^2+2^2+3^2+4^2+...............+n^2= \frac{n(n+1)(2n+1)}{6} }$
see pattern in
12+ 22+ 32+ 42 + ... + 252= 5525
LHS =1*1+2*2+3*3+4*4+............+25*25
$=1^2+2^2+3^2+4^2+...............+25^2$
$=\frac{25(25+1)(50+1)}{6}=5525$= RHS

therefore
22+ 42+ 62 + 82 + ... + 502 = N

$2^2+4^2+6^2+8^2+10^+.....+50^2$
$=4+16+36+64+100+.......+2500$
$=4(1+4+9+16+25+.........+625)$
$=4(1^2+2^2+3^2+4^2+...............+25^2)$
$= 4 \cdot \frac{25(25+1)(50+1)}{6} =22100$

3. Many thanks for taking the time to answer ! Appreciated.