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If the sum of four numbers in AP be 96 and that of the product of the means to the product of the extremes is 35:27, then find the numbers.

+3 votes
296 views
asked Aug 20, 2016 in Mathematics by Rahul Roy (7,955 points)

If the sum of four numbers in AP be 96 and that of the product of the means to the product of the extremes is 35:27, then find the numbers.

1 Answer

+5 votes
answered Aug 20, 2016 by vikash (21,287 points)
selected Sep 22, 2016 by sarthaks
 
Best answer

Take the four numbers as (a-3d),(a-d),(a+d),(a+3d) in AP with common difference as 2d.

Now given that Sum of four numbers = 96

i.e (a-3d)+(a-d)+(a+d)+(a+3d) = 96

=> 4a = 96

=> a = 24 ----------(1)

Now to find d use the other condition.

i.e Product of means/Product of Extremes = 35/27

=> (a-d)(a+d) / (a-3d)(a+3d) = 35/27

=> a- d2 / a2 -9d2 = 35/27 ---------------(2)

Now find the value of d using equation (1) and (2)

You will get d = 4

So the four numbers are:

a-3d = 24-3*4 = 12

a-d = 24 - 4 = 20

a+d = 24+4 = 28

a+3d = 24+12 = 36

Hence the four numbers are 12,20,28,36

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