Posted by Meme J on Wednesday, September 5, 2012 at 12:39am.
5 = 1 + 4
6 = 2 + 4
7 = 3 + 4
...
14 = 10 + 4
5 ^ 2 + 6 ^ 2 + 7 ^ 2 + ... + 14 ^ 2 =
( 1 + 4 ) ^ 2 + ( 2 + 4 ) ^ 2 + ( 3 + 4 ) ^ 2 + ... + ( 10 + 4 )^ 2 =
E ( k + 4 ) ^ 2 from n = 1 to n = 10
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( k + 4 ) ^ 2 = k ^ + 2 * k * 4 + 4 ^ 2 =
k ^ 2 + 8 k + 16
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E ( k + 4 ) ^ 2 from n = 1 to n = 10 =
E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + E 16 from n = 1 to n = 10 =
E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + 10 * 16 =
E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + 160
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E k ^ 2 from n = 1 to n = 10
= n * ( n + 1 ) * ( 2 n + 1 ) / 6 =
10 * ( 10 + 1 ) * ( 2 * 10 + 1 ) / 6 =
10 * 11 * 21 / 6 = 2310 / 6 = 385
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E k from n = 1 to n = 10
= n * ( n + 1 ) / 2 = 10 * 11 / 2 = 110 / 2 = 55
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E ( k + 4 ) ^ 2 from n = 1 to n = 10
= E k ^ 2 from n = 1 to n = 10 + 8 * E k from n = 1 to n = 10 + 160
= 385 + 8 * 55 + 160 = 385 + 440 + 160 = 985