標題:
What is the sum of the square of the first 50positive even..
發問:
The sum of the squares of the first fifty positive integers is 42,925. What is the sum of the square of the first fifty positive even integers? the answer is 171700 i know the answer, but i need the steps show clearly, and explain as long as u can. Like, formula, sequence, or other stuff... 更新: Explain in words plz better explain in english..but chinese is also fine Thanks ^^
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 //
其他解答:
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)
What is the sum of the square of the first 50positive even..
發問:
The sum of the squares of the first fifty positive integers is 42,925. What is the sum of the square of the first fifty positive even integers? the answer is 171700 i know the answer, but i need the steps show clearly, and explain as long as u can. Like, formula, sequence, or other stuff... 更新: Explain in words plz better explain in english..but chinese is also fine Thanks ^^
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最佳解答: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 //
其他解答:
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)
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