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
What does AP mean and what does this have to do with Christ?
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
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
To find the arithmetic progression (AP) that satisfies the given conditions, we can follow these steps:
Step 1: Let's assume that the four parts of 20 in the AP are a-d, a, a+d, and a+2d. Here, 'a' represents the first term of the AP, and 'd' represents the common difference of the AP.
Step 2: Now, let's calculate the product of the first part and the fourth part, which is (a-d) * (a+2d), and the product of the second and third part, which is a * (a+d).
Step 3: According to the given condition, the ratio between the product of the first and fourth part to the second and third part is 2:3. This can be expressed as:
(a-d) * (a+2d) : a * (a+d) = 2 : 3
Step 4: Simplifying the above equation, we get:
(a^2 + 2ad - ad - 2d^2) : (a^2 + ad) = 2 : 3
Step 5: Cross-multiplying the ratios, we have:
3(a^2 + 2ad - ad - 2d^2) = 2(a^2 + ad)
Step 6: Expanding the equation and simplifying, we get:
3a^2 + 6ad - 3ad - 6d^2 = 2a^2 + 2ad
Step 7: Further simplifying the equation, we have:
(3a^2 - 2a^2) + (6ad - 3ad) - 6d^2 = 0
Step 8: Combining like terms, we get:
a^2 + 3ad - 6d^2 = 0
Step 9: Now, let's solve this quadratic equation for 'a' using factoring or the quadratic formula. Assuming that the solution for 'a' is 'm' and 'n', we get:
(a - m)(a - n) = 0
Step 10: Equating each factor to zero, we have:
a - m = 0 or a - n = 0
Therefore, the solutions for 'a' are 'm' and 'n'. Once we find the values for 'a', we can find the common difference 'd' using the formula:
d = (2nd term - 1st term)
With this information, we can now determine the arithmetic progression (AP) that divides 20 into four parts.