# Good Probs - Question The sum of the squares of the first...

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08/11/03 Question The sum of the squares of the first 15 positive integers (1 2 + 2 2 + 3 2 + . . . + 15 2 ) is equal to 1240. What is the sum of the squares of the second 15 positive integers (16 2 + 17 2 + 18 2 + . . . + 30 2 ) ? (A) 2480 (B) 3490 (C) 6785 (D) 8215 (E) 9255 Answer The key to solving this problem is to represent the sum of the squares of the second 15 integers as follows: (15 + 1) 2 + (15 + 2) 2 + (15 + 3) 2 + . . . + (15 + 15) 2 . Recall the popular quadratic form, (a + b) 2 = a 2 + 2ab + b 2 . Construct a table that uses this expansion to calculate each component of each term in the series as follows: (a + b) 2 a 2 2ab b 2 (15 + 1) 2 225 2(15)1 =30 1 2 (15 + 2) 2 225 2(15)2 =60 2 2 (15 + 3) 2 225 2(15)3 =90 3 2 . . . . . . . . . . . . (15 + 15) 2 225 2(15)15 =450 15 2 TOTALS = 15(225)=3375 (30+450)/2 × 15 = 3600 1240 In order to calculate the desired sum, we can find the sum of each of the last 3 columns and then add these three subtotals together. Note that since each column follows a simple pattern, we do not have to
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