What is the sum of the square of the first 50positive even..

since we have :

1^2 + 2^2 + 3^2 + ......+ 50^2 =42,925

now, we have first fifty positive even intergers, that is

2^2 + 4^2 + 6^2+................+ (2 x 50)^2
= (2x1)^2 + (2x2)^2 + (2x3)^2 .+..............+ (2x50)^2
=(2^2)(1^2) + (2^2)(2^2) +(2^2)(3^2)+..........+(2^2)(50^2)
= 2^2( 1 + 2^2 + 3^2 +.....................+ 50^2)
=4 (42,925)
=171700 //

2007-11-23 16:54:24 補充：
題目給予由 1的次方加到 50的次方如下:1^2  2^2  3^2  ...... 50^2 =42,925

2007-11-23 16:56:23 補充：
題目給予由 1的次方加到 50的次方如下:1^2 ┼ 2^2 ┼ 3^2┼  .....┼. 50^2 =42,925

2007-11-23 16:59:15 補充：
現在利用上面的等式, 求開頭 50 個正偶數的次方, 即是: 2^2┼ 4^2 ┼ 6^2┼...............┼ (2 x 50)^2 = (2x1)^2 ┼ (2x2)^2 ┼ (2x3)^2 ┼...............┼ (2x50)^2=(2^2)(1^2) ┼ (2^2)(2^2) ┼(2^2)(3^2)┼.........┼.(2^2)(50^2)= 2^2( 1 ┼ 2^2 ┼ 3^2 ┼.....................┼ 50^2) =4 (42,925)=171700 //

since you want a sum of a square of square even number, it will be same as saying
2 square (whcih is 4) times the sum of the n square from n is 1 to 50


the sum of the n^2 formula is n(n+1)(2n+1)/6

so here n = 50

ans = 4 x 50(50+1)(100+1)/6 = (200 x 51 x 101)/6
= (2 x 100 x 3 x 17 x 101 )/6
= 17x101x 100
= 17 x (100+1) x100
= (1700 +17) x 100
= 171700

(you can use your caculator from the 1st line, the following is just my stupid method)

2007-11-22 23:55:49 補充：
sorry first bit actually not quite clear sum of the (2n)^2 = (2^2) x (n^2) (n is a integer from 1 to 50) = 4 n^2(this insert of the first 2 line of my previous work will be easy to understand)

2² + 4² + ... + 100²

= 2² × (1² + 2² + ... + 50²)

= 2² × (1/6) × (50) × (50 + 1) × (2 × 50 + 1)

= 4 × (1/6) × (50) × (51) × (101)

= 171700

Note: 1² + 2² + ... + 50² = (1/6) × n × (n + 1) × (2n + 1)