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# The first term, the mean, the last term, and the sum of an arithmetic

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VP
Joined: 07 Dec 2014
Posts: 1252
The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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16 Mar 2019, 09:48
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Difficulty:

75% (hard)

Question Stats:

59% (02:52) correct 41% (02:49) wrong based on 73 sessions

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The first term, the mean, the last term, and the sum of an arithmetic progression of nine terms are all perfect squares. What is the sum of these four squares?

A. 289
B. 300
C. 376
D. 415
E. 487
VP
Joined: 07 Dec 2014
Posts: 1252
The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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16 Mar 2019, 15:40
4
2
jfranciscocuencag wrote:
gracie

Hello!

Regards!

hi,

let x=first term
y=last term
sum of four squares=x+y+(x+y)/2+[(x+y)/2]9➡
6(x+y)
300 is only multiple of 6
x=1, y=49, mean=25, sum=225
1+49+25+225=300
B

I hope this helps.
##### General Discussion
Senior Manager
Joined: 12 Sep 2017
Posts: 306
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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16 Mar 2019, 13:13
1
gracie

Hello!

Regards!
Intern
Joined: 18 Oct 2017
Posts: 1
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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17 Mar 2019, 19:14
Hi Grace,

Can you please explain why did you multiply 9 with the mean?

Regards,
JM
VP
Joined: 07 Dec 2014
Posts: 1252
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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19 Mar 2019, 10:08
JimitMehta wrote:
Hi Grace,

Can you please explain why did you multiply 9 with the mean?

Regards,
JM

hi, JM

because there are 9 terms in the sequence.
9 terms*mean (25)=sum of total sequence (225)

I hope this helps,
gracie
Intern
Joined: 12 Feb 2018
Posts: 12
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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26 Mar 2019, 20:03
4
Let the sequence be a-4d, a-3d, a-2d, a-d, a, a+d, a+2d, a+3d, a+4d

First term + Last term + mean + sum of terms = a-4d + a+4d + a + 9a = 12a

The only term divisible by 12 is 300
Intern
Joined: 19 Nov 2017
Posts: 3
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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27 Mar 2019, 01:18
I solved it this way.

take the first number as 6, 2nd number as 10, then the mean becomes 8 [ Essentially the 3-4-5 triangle rule as only that can be an equally spaced set of squares ]
the addition of the squares of all three gives 200. subtracting it from the answer choices given, Only 300 yields a perfect square i.e. 100
Ans. B
Senior Manager
Joined: 12 Sep 2017
Posts: 306
The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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12 Apr 2019, 13:37
1
gracie wrote:
jfranciscocuencag wrote:
gracie

Hello!

Regards!

hi,

let x=first term
y=last term
sum of four squares=x+y+(x+y)/2+[(x+y)/2]9➡
6(x+y)
300 is only multiple of 6
x=1, y=49, mean=25, sum=225
1+49+25+225=300
B

I hope this helps.

Hello gracie!

How could it be 49 the last term if its an arithmetic progression of 9 terms?

Shouldn't be:

nth = a + d(n-1)

1(a),4,9,16,25(mean),36,49,64,81... 9 terms in total.

So

a = 1
nth = 81
Mean = 25
Sum of terms = 285

Mmmm now that I am writing it I guess is wrong cuz it should be a geometric progression, isn't it?
Manager
Joined: 09 Jun 2017
Posts: 102
GMAT 1: 640 Q44 V35
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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14 Apr 2019, 02:15
guys , let the arithmetic progression of nine terms be x
so what is the sum of an arithmetic progression of nine terms ? 8x or 9x ?
CEO
Joined: 03 Jun 2019
Posts: 2910
Location: India
GMAT 1: 690 Q50 V34
WE: Engineering (Transportation)
The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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18 May 2020, 09:17
gracie wrote:
The first term, the mean, the last term, and the sum of an arithmetic progression of nine terms are all perfect squares. What is the sum of these four squares?

A. 289
B. 300
C. 376
D. 415
E. 487

Given: The first term, the mean, the last term, and the sum of an arithmetic progression of nine terms are all perfect squares.
Asked: What is the sum of these four squares?

Let a be first term, d be common difference and n be number of terms of an arithmetic progression

First term a is a perfect square
Mean = (l+a)/2 is a perfect square
Last term l = a + (n-1)d; is a perfect square
Sum of 9 terms = (9/2) (l+a); is a perfect square

a + (l+a)/2 + l + 9(l+a)/2 = 6a + 6l = 6(a+l)
300 is the only multiple of 6
a + l = 50;

Let a be 1 and l be 49; (n-1)d = 48;
Mean = 25
Sum of 9 terms = 9*25 = 225
6(a+l) = 6*50 = 300

IMO B
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Kinshook Chaturvedi
Email: kinshook.chaturvedi@gmail.com
The first term, the mean, the last term, and the sum of an arithmetic   [#permalink] 18 May 2020, 09:17