Four terms in arithmetic progression have sum of 28. Product

Question

Four terms in arithmetic progression have sum of 28. Product of the extreme terms and that of the middle terms are in the ratio 5:6. Find the largest term?

A. 8
B. 9
C. 10
D. 11
E. 12

KarishmaB · Accepted Answer

Using brute force can be quite helpful.

Sum of 28 means that essentially all 4 numbers are equal to 7 each. Since the product of 1st and 4th is different from the product of 2nd and 3rd terms, the common difference must be more than 0.
So the numbers could be 4, 6, 8, 10 (with 7 as the mean) when the common difference is more than 0 but least possible (assuming integers)
Product of 1st and 4th = 40
Product of 2nd and 3rd = 48
Ratio = 5:6 (what we wanted)
So the largest term is 10.

Answer (C)

Had we not obtained the required ratio, we would have increased the common difference and taken the terms as 1, 5, 9, 11 and then checked. This would have given us a pattern to tell us whether the ratio is increasing or decreasing.

kinjiGC · Answer

lets take the terms as (a-3d), (a-d), (a+d), (a+3d).
as sum = 28, so a = 7

Lets check the option and we can see 7+3d can be equal to 10. lets test that scenario.

So the numbers can be 4,6,8,10. product of extreme terms/product of middle terms = 40/48 = 5/6 So Option C)

We can use this technique to solve GMAT problems as none of the above or cannot be determined is yet to be an option in GMAT.

For the Maths purists :

(a-3d) * (a+3d) /(a-d) * (a+d) = 5/6 => 6(a^2 - 9d^2) = 5(a^2 - d^2) => a^2 = 49d^2 as a = 7 then d has to be 1. hence option c)

As we don't get marks for steps I prefer the approach 1.

sagnik242 · Answer

Lets check the option and we can see 7+3d can be equal to 10. lets test that scenario.

So the numbers can be 4,6,8,10. product of extreme terms/product of middle terms = 40/48 = 5/6 So Option C)

Question : how did you know 7 + 3d can be equal to 10?

How did you get #'s 4,6, 8, 10?

kinjiGC · Answer

When using the options , start with option C). Normally the options will be ordered :-D

(How I know it, I also develop question items ) if option C) doesn't satisfy the equation, at least you can have a range of numbers which might satisfy and you can eliminate the options easily. Coming to the question: if 7+3d = 10, so we can calculate the value of d= 1 and we can calculate the other three numbers. as 1st term is a-3d, so putting a =7 and d =1, we can get 4 and so on. Any doubts, please revert.

bankerboy30 · Answer

Why did you set up the equations (a-3d), (a-d), ect

kinjiGC · Answer

That is simply as assumption. It reduces the calculation time and complexity.

If we have three terms in AP, we can assume
a-d, a, a+d

If we have four terms in AP, we can assume
a-3d, a-d, a+d, a+3d

if we have 5 terms in AP, we can assume
a-2d, a-d, a, a+d, a+2d

So if as we add the terms, the terms cancel each other and we can have some advantage in calculation.

gracie · Answer

let a=1st term and d=difference between terms
4a+6d=28→2a+3d=14
2 possibilities: a=1; d=4 and a=4; d=2
but only 4 and 2 give 5:6 extreme:middle term product ratio
so 10
C

clefebvre3 · Answer

30 seconds method

W+X+Y+Z = 28

To get X and Y

Mean: 28/4 = 7 
Average of 7: (6+8)/2
 
X= 6 Y=7 (closest to the mean)

To get W,Z

Given, Ratio 5:6
WZ:48 = 5:6
WZ= 40

W= 4 ; Z= 10

vitorcmonteiro · Answer

I have used the following steps:

1 - Set  A, A+n, A+2n, A+3n , SUM OF SET is  4A+6n = 28 
2 - Simplify to  2A+3n = 14 
3 -  A + A + 3n = 14  and thus  A + 3n = 14 - A  (Meaning the largest is 14 minus smallest)
4 - Started picking numbers from C (POE strategy),  A + 3n = 10  (Largest is 10),  14 - A = 10  thus A should be 4.
5 -  4 + 3n = 10  >>>  n = 2 
6 -  \frac{(10 * 4)}{(6 * 8)}  =  \frac{40}{48}  =  \frac{5}{6}

May take more than a minute to solve, however. But I hope it could still help to understand the logic behind. Please correct me if I am wrong!

TheNightKing · Answer

Well, I was trying to solve the numbers but then realised something. If the product of extreme terms is in the form of 5x that means one of them has to be a 5 or a multiple of 5. 
The only option is C which is 10.
Is that fair?

Bunuel

Bunuel · Answer

____________________
Seems fair to me.

TheNightKing · Answer

Thank you!

Archit3110 · Answer

the AP of the given series of no would be of possiblity 4,6,8,10 
4*10/6*8 = 5:6
IMO C ; 10

Kinshook · Answer

Given: 
1. Four terms in arithmetic progression have sum of 28. 
2. Product of the extreme terms and that of the middle terms are in the ratio 5:6.

Asked: Find the largest term?

Let the 4 terms in AP be {a-3d, a-d , a+d, a+3d}
4a = 28; a = 7
 \frac{(7-3d)(7+3d)}{(7-d)(7+d)} = \frac{5}{6} 
 \frac{49 - 9d^2}{49 - d^2} = \frac{5}{6} 
 \frac{49 - 9d^2}{49 - d^2} = \frac{5}{6} 
6(49 - 9d^2) = 5(49-d^2)
49 = 54d^2 - 5d^2 = 49d^2
d = +-1

Terms = {4,6,8,10}
Largest term = 10

IMO C

MathRevolution · Answer

Let the four numbers in the A.P. are (a - 2d), (a - d), (a + d) (a + 2d)

Sum: (a - 2d) + (a - d) + (a + d) + (a + 2d) = 28

=> 4a = 28 and hence a = 7

If we take numbers to be  : 7 * 10 is 70 and 8 * 9 = 72.

=> 70: 72 is 5: 6 and the hence largest term is 10.

Answer C

prabgupta · Answer

lets use options to solve this question.
since it is an AP, and sum = 28, the sum of middle terms = 14 and sum of extreme must also be 14.
a1 + a4 = 14 and a2 + a3 = 14.
now use options.
A--> if a4 =8, then a1 = 6, now make an AP from it, a1 =6, a4 = a1+3d = 8, d = 2/3, check ratio it would not be equal to 5/6.
B--> if a =9, then a = 5, check again
C--> if a =10, then d =4, very intuitive, 4, 6, 8, 10--> Done
D

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Four terms in arithmetic progression have sum of 28. Product

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