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25 Nov 2019, 03:54
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
31% (02:57) correct
69%
(02:30)
wrong
based on 251
sessions
History
Date
Time
Result
Not Attempted Yet
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
(A) 2485
(B) 2585
(C) 2685
(D) 2785
(E) 2895
25 Nov 2019, 06:15
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CaptainLevi
\(S=(a-5)^2+(a-4)^2+(a-3)^2+(a-2)^2+(a-1)^2+a^2+(a+1)^2+(a+2)^2+(a+3)^2+(a+4)^2\)
=\((a-5)^2+(a-4)^2+(a+4)^2+...=a^2-10a+25+2(a^2+4^2)+2(a^2+3^2)..+2(a^2).\)
=\(a^2-10a+25+a^2+2(4a^2+1^2+2^2+3^2+4^2)=a^2-10a+25+a^2+2(4a^2+1+4+9+16)\)
=\(10a^2-10a+25+60=10a(a-1)+85\)
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
General Discussion
alamin8
Joined: 24 May 2025
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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.
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.
