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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # The first term, the mean, the last term, and the sum of an arithmetic

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VP  D
Joined: 07 Dec 2014
Posts: 1259
The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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11 00:00

Difficulty:   75% (hard)

Question Stats: 59% (02:47) correct 41% (02:46) wrong based on 76 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  D
Joined: 07 Dec 2014
Posts: 1259
The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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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
Intern  B
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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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
Senior Manager  G
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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1
gracie

Hello!

Regards!
Senior Manager  G
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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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?
Intern  B
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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Hi Grace,

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

Regards,
JM
VP  D
Joined: 07 Dec 2014
Posts: 1259
Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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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  B
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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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
Manager  B
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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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  V
Joined: 03 Jun 2019
Posts: 3217
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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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
_________________
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

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