The sum of the squares of the first 15 positive integers GMAT Problem Solving

Question: The sum of the squares of the first 15 positive integers (1² + 2^{2 +} 3² + … + 15²) is equal to 1240. What is the sum of the squares of the second 15 positive integers (16² +17² + 18² +… + 30²)?

     A. 2480
     B. 3490
     C. 6785
     D. 8215
     E. 9255

Answer: D

Approach Solution (1):
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 = a²+ 2ab + b². Construct a table that uses this expansion to calculate each component of each term in the series as follows:

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 fill in the whole table, but instead only need to calculate a few terms in order to determine the sums.
The column labeled a² simply repeats 225 fifteen times; therefore, its sum is 15(225) = 3375.
The column labeled 2ab is an equally spaced series of positive numbers. Recall that the average of such a series is equal to the average of its highest and lowest values; thus, the average term in this series is (30 + 450) / 2 = 240. Since the sum of n numbers in an equally spaced series is simply n times the average of the series, the sum of this series is 15(240) = 3600.
The last column labeled b² is the sum of the squares of the first 15 integers. This was given to us in the problem as 1240.
Finally, we sum the 3 column totals together to find the sum of the squares of the second 15 integers: 3375 + 3600 + 1240 = 8215
Correct option: D

Approach Solution (2):
Sum of the square of n numbers is found using the formula: [n(n+1)(2n+1)] / 6
We have sum of 1^st 15 numbers = 1240
We need to find the sum of squares from 16 to 30 which is = sum of squares of 1^st 30 +ve integers = sum of the squares of 1^st 15 +ve integers
We will get: 8215
Correct option: D

Approach Solution (3):
Sum of the squares is [n(n+1)(2n+1)] / 6 (for 1^st n natural numbers)
So for 1st 15 – (15*16*31) / 6 = 5*8*31 = 1240
For 1st 30 – (30*31*61) / 6 = 5*61*31 = 9455
9455 – 1240 = 8215
Correct option: D

“The sum of the squares of the first 15 positive integers (1² + 2^{2 +} 3² + … + 15²) is equal to 1240. What is the sum of the squares of the second 15 positive integers (16² +17² + 18² +… + 30²)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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