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Joined: 16 May 2014
Posts: 592
Given Kudos: 0
GRE 1: Q165 V161
The sum of the squares of the first 15 positive integers
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12 May 2016, 00:33
12 May 2016, 00:33
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Question Stats:
50% (03:12) correct
50% (01:28) wrong based on 10 sessions
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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
Kudos to the best and the quickest explanation!
(A) 2480
(B) 3490
(C) 6785
(D) 8215
(E) 9255
Kudos to the best and the quickest explanation!
Re: The sum of the squares of the first 15 positive integers
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13 May 2016, 00:29
13 May 2016, 00:29
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8215
Explanation:-
Sum of square of first n natural numbers is given by (n(n+1)(2n+1))/6
So taking the sum of square of first 30 natural numbers and then subtracting the sum of the square of first 15 natural numbers that is already given in the question
Explanation:-
Sum of square of first n natural numbers is given by (n(n+1)(2n+1))/6
So taking the sum of square of first 30 natural numbers and then subtracting the sum of the square of first 15 natural numbers that is already given in the question
Re: The sum of the squares of the first 15 positive integers
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15 May 2016, 20:07
15 May 2016, 20:07
2
Can also be done without the formula
(16^2 + 17^2 + 18^2 + . . . + 30^2) - (1^2 + 2^2 + 3^2 + . . . + 15^2) = x - 1240
= (16^2 - 1^2) + (17^2 - 2^2) + ...+ (30^2 - 15^2)
= (16 - 1)(16 + 1) + ....+(30 - 15)(30 + 15)
= 15 * 17 + 15 * 19 + .........+ 15 * 45
= 15 (17 + 19 + ....+ 45) = 15 * 31 * 15 (31 being the mean on 17, 19, ... 45)
= 6975 = x - 1240 => x = 6975 + 1240 = 8215
(16^2 + 17^2 + 18^2 + . . . + 30^2) - (1^2 + 2^2 + 3^2 + . . . + 15^2) = x - 1240
= (16^2 - 1^2) + (17^2 - 2^2) + ...+ (30^2 - 15^2)
= (16 - 1)(16 + 1) + ....+(30 - 15)(30 + 15)
= 15 * 17 + 15 * 19 + .........+ 15 * 45
= 15 (17 + 19 + ....+ 45) = 15 * 31 * 15 (31 being the mean on 17, 19, ... 45)
= 6975 = x - 1240 => x = 6975 + 1240 = 8215
