Posted by **pavithra** on Tuesday, March 12, 2013 at 7:01am.

Divide 20 into 4 parts which are in AP such that ratio between the product of the 1st part and the 4th part to the 2nd and 3rd part is 2:3 find the AP

- CHRIST MATH -
**drwls**, Tuesday, March 12, 2013 at 7:04am
What does AP mean and what does this have to do with Christ?

- CHRIST MATH -
**Reiny**, Tuesday, March 12, 2013 at 9:50am
I will assume that AP stands for arithmetic progression

so you are given:

a + (a+d) + (a+2d) + (a+3d) = 20

4a + 6d = 20

2a + 3d = 10

product of 1st and 4th = a(a+3d)

product of 2nd to 3rd = (a+d)(a+2d)

then:

a(a+3d)/( (a+d)(a+2d) ) = 2/3

(a^2 + 3ad)/(a^2 + 3ad + 2d^2) = 2/3

3a^2 + 9ad = 2a^2 + 6ad + 4d^2 = 0

a^2 + 3ad - 4d^2 = 0

(a-d)(a+4d) = 0

a = d or a = -4d

case1: a = d

in 2a+3d=10

5d = 10,

d = 2, so a=d = 2

**the terms are 2, 4, 6 , 8**

case2: a = 4d

in 2a + 3d = 10

-4d + 3d = 10

-d = 10

d = -10, then a = 40

**the terms are 40, 30, 20, 10**

- typo - CHRIST MATH -
**Reiny**, Tuesday, March 12, 2013 at 10:00am
I just noticed a typo in my second case

It should say:

case2: a = -4d

in 2a + 3d = 10

-8d + 3d = 10

-2d = 10

d = -5, then a = 20

the terms are 20 , 15, 10 , 5

but these do not add up to 20 , even though the ratio of product property works

so the only solution are the terms

2, 4, 6, 8

notice they add up to 20

and (2x8)/(4x6) = 16/24 = 2/3

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