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# Let S denote the sum of the squares of ten consecutive positive

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Manager
Joined: 18 Jul 2019
Posts: 54
Let S denote the sum of the squares of ten consecutive positive  [#permalink]

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25 Nov 2019, 03:54
1
5
00:00

Difficulty:

95% (hard)

Question Stats:

10% (02:19) correct 90% (02:14) wrong based on 21 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
Math Expert
Joined: 02 Aug 2009
Posts: 8284
Re: Let S denote the sum of the squares of ten consecutive positive  [#permalink]

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25 Nov 2019, 06:15
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1
CaptainLevi wrote:
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

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

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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Joined: 24 Nov 2019
Posts: 3
Re: Let S denote the sum of the squares of ten consecutive positive  [#permalink]

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30 Nov 2019, 21:58
CaptainLevi wrote:
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

Could you please share the solution?
Re: Let S denote the sum of the squares of ten consecutive positive   [#permalink] 30 Nov 2019, 21:58
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