Q.200. Arithmetical Progression

Question 200.
If we divide 20 into four parts which are in A.P. such that the ratio of the product of the first and the fourth term to the product of the 2nd and 3rd terms is 2/3, then find the smallest term.

Answer 200.
Let the four numbers in A.P. be

a - 3d, a - d, a + d, a + 3d
=> 4a = 20
=> a = 5

Also,
(a^2 - 9d^2) / (a^2 - d^2) = 2/3
=> 3a^2 - 27d^2 = 2a^2 - 2d^2
=> 25d^2 = a^2 = 25
=> d = 1 [d = - 1 can also be considered, but will yield the same result]
=> Smallest part is
a - 3d
= 5 - 3
= 2. ... [the next three numbers are 4, 6, 8]

Answer: 2.

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