The first term, the mean, the last term, and the sum of an arithmetic : Problem Solving (PS)

P

gracie

VP

Joined: 07 Dec 2014

Posts: 1075

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

16 Mar 2019, 10:48

25

Bookmarks

00:00

A

B

C

D

E

Difficulty:

85% (hard)

Question Stats:

57% (02:52) correct

43% (02:54) wrong

based on 189 sessions

Hide Show timer Statistics

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

Show Answer

Official Answer

Official Answer and Stats are available only to registered users. Register/Login.

P

gracie

VP

Joined: 07 Dec 2014

Posts: 1075

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

16 Mar 2019, 16:40

6

Kudos

3

Bookmarks

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.

B

poojachaudhary0305

Intern

Joined: 12 Feb 2018

Posts: 9

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

26 Mar 2019, 21:03

6

Kudos

3

Bookmarks

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

G

jfranciscocuencag

Manager

Joined: 12 Sep 2017

Posts: 239

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

16 Mar 2019, 14:13

1

Kudos

gracie

Hello!

Could you please provide an answer?

Regards!

G

jfranciscocuencag

Manager

Joined: 12 Sep 2017

Posts: 239

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

12 Apr 2019, 14:37

1

Kudos

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?

B

JimitMehta

Intern

Joined: 18 Oct 2017

Posts: 1

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

17 Mar 2019, 20:14

Hi Grace,

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

Regards,
JM

P

gracie

VP

Joined: 07 Dec 2014

Posts: 1075

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

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

B

Jayvnandu1991

Intern

Joined: 20 Nov 2017

Posts: 2

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

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

B

foryearss

Manager

Joined: 09 Jun 2017

Posts: 86

GMAT 1: 640 Q44 V35

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

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 ?

L

Kinshook

GMAT Club Legend

Joined: 03 Jun 2019

Posts: 5345

Location: India

Schools: HBS '22 CBS '22 Wharton '22

GMAT 1: 690 Q50 V34 Verified score

WE:Engineering (Transportation)

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

18 May 2020, 10:17

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

L

Archit3110

GMAT Club Legend

Joined: 18 Aug 2017

Status:You learn more from failure than from success.

Posts: 8026

Location: India

Concentration: Sustainability, Marketing

GMAT Focus 1:
545 Q79 V79 DI73 Verified score

GPA: 4

WE:Marketing (Energy and Utilities)

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

28 Jul 2021, 08:06

sum of squares = 1/6 * ( n) *( n+1)*(2n+1)
n=4
sum of squares = 1/6 *4*5*9
sum of squares = 30
so answer option has to be multiple of 30
which is only option B 300

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

L

Fdambro294

VP

Joined: 10 Jul 2019

Posts: 1394

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

29 Jul 2021, 11:55

When you use the Sum of Perfect Squares formula, aren’t you taking the Sum of the 1st N perfect Squares?

Thus, the equation took the sum of:

(1)^2 + (2)^2 + (3)^2 + (4)^2 = 30

Does that logically get us to the right answer? Just want to check to see if I’m missing something

Thank you much for any help!

Archit3110 wrote:

sum of squares = 1/6 * ( n) *( n+1)*(2n+1)
n=4
sum of squares = 1/6 *4*5*9
sum of squares = 30
so answer option has to be multiple of 30
which is only option B 300

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

Posted from my mobile device

bumpbot

Non-Human User

Joined: 09 Sep 2013

Posts: 32485

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

03 Nov 2023, 05:40

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Signature Read More

gmatclubot

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

03 Nov 2023, 05:40

GMAT Problem Solving (PS) Questions

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

GMAT Question Banks