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CaptainLevi
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Let S denote the sum of the squares of ten consecutive positive 25 Nov 2019, 03:54
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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95% (hard)

Question Stats:

32% (02:58) correct 68% (02:37) wrong based on 226 sessions

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Let S denote the sum of the squares of ten consecutive positive integers. Which of the following values represents a possible value of S?

(A) 2485
(B) 2585
(C) 2685
(D) 2785
(E) 2895
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A
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chetan2u
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Let S denote the sum of the squares of ten consecutive positive 25 Nov 2019, 06:15
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CaptainLevi
Let S denote the sum of the squares of ten consecutive positive integers. Which of the following values represents a possible value of S?

(A) 2485
(B) 2585
(C) 2685
(D) 2785
(E) 2895
Show more

\(S=(a-5)^2+(a-4)^2+(a-3)^2+(a-2)^2+(a-1)^2+a^2+(a+1)^2+(a+2)^2+(a+3)^2+(a+4)^2\)
=\((a-5)^2+(a-4)^2+(a+4)^2+...=a^2-10a+25+2(a^2+4^2)+2(a^2+3^2)..+2(a^2).\)
=\(a^2-10a+25+a^2+2(4a^2+1^2+2^2+3^2+4^2)=a^2-10a+25+a^2+2(4a^2+1+4+9+16)\)
=\(10a^2-10a+25+60=10a(a-1)+85\)

Check the choices..

(A) 2485......10a(a-1)+85=2485......10a(a-1)=2400......a(a-1)=240=15*16...Possible
(B) 2585......10a(a-1)+85=2585......10a(a-1)=2500......a(a-1)=250=2*5^3...Not Possible
Similarly for other choices

A
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Let S denote the sum of the squares of ten consecutive positive 30 Jul 2026, 17:43
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Let the smallest number is "n". Then the consecutive positive integer will be n + (n+1) + (n+2)+ .... (n+9)
Therefore, sum of square = n^2 + (n+1)^2 + .... (n+9)^2 = 10n^2 + 90n + 285

There are 2 ways we can do. Either by plugging a number or by taking equations
Process 1:
since the answer choices are big,
n = 10, 10x100 + 90x10 + 285 = 1000 + 900 + 285 = 2185
n = 11, 10x121 + 90x11 + 285 = 1210 + 990 + 285 = 2485 (Option A)
n = 12, 10x144 + 90x12 + 285 = 1440 + 1080 + 285 = 2805 (Option B, C, D, E out)

Therefore, Answer is A

Process 2:
10n^2 + 90n + 285 = 2485
10n^2 + 90n - 2200 = 0
n^2 + 9n - 220 = 0
n^2 + 20n - 11n - 220 = 0
(n + 20) (n - 11) = 0
Therefore, n = 11
We get a value! It is the answer

If we go with option B
10n^2 + 90n + 285 = 2585
10n^2 + 90n - 2300 = 0
n^2 + 9n - 230 = 0

Not possible to break it down.

Therefore, the answer is A.
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Let S denote the sum of the squares of ten consecutive positive 31 Jul 2026, 23:36
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awesome, I tried this as well
alamin8
Let the smallest number is "n". Then the consecutive positive integer will be n + (n+1) + (n+2)+ .... (n+9)
Therefore, sum of square = n^2 + (n+1)^2 + .... (n+9)^2 = 10n^2 + 90n + 285

There are 2 ways we can do. Either by plugging a number or by taking equations
Process 1:
since the answer choices are big,
n = 10, 10x100 + 90x10 + 285 = 1000 + 900 + 285 = 2185
n = 11, 10x121 + 90x11 + 285 = 1210 + 990 + 285 = 2485 (Option A)
n = 12, 10x144 + 90x12 + 285 = 1440 + 1080 + 285 = 2805 (Option B, C, D, E out)

Therefore, Answer is A

Process 2:
10n^2 + 90n + 285 = 2485
10n^2 + 90n - 2200 = 0
n^2 + 9n - 220 = 0
n^2 + 20n - 11n - 220 = 0
(n + 20) (n - 11) = 0
Therefore, n = 11
We get a value! It is the answer

If we go with option B
10n^2 + 90n + 285 = 2585
10n^2 + 90n - 2300 = 0
n^2 + 9n - 230 = 0

Not possible to break it down.

Therefore, the answer is A.
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