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

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

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New post 16 Mar 2019, 10:48
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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
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The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 16 Mar 2019, 16:40
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jfranciscocuencag wrote:
gracie

Hello!

Could you please provide an answer?

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.
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Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 16 Mar 2019, 14:13
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gracie

Hello!

Could you please provide an answer?

Regards!
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Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 17 Mar 2019, 20:14
Hi Grace,

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

Regards,
JM
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Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 19 Mar 2019, 11:08
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
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Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 26 Mar 2019, 21:03
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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
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Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 27 Mar 2019, 02:18
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
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The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 12 Apr 2019, 14:37
gracie wrote:
jfranciscocuencag wrote:
gracie

Hello!

Could you please provide an answer?

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!

Could you please help me with the following?

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?
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Re: The first term, the mean, the last term, and the sum of an arithmetic  [#permalink]

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New post 14 Apr 2019, 03:15
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 ?
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Re: The first term, the mean, the last term, and the sum of an arithmetic   [#permalink] 14 Apr 2019, 03:15
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