An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP.

Given: Sum of three middle terms = 225

a₃₅ + a₃₆ + a₃₇ = 429

To find: The AP.

Formula Used:

Sum of “n” terms of an AP:

Where S_n is the sum of first n terms

nth term of an AP is:

a_{n =} a + (n - 1) d

Where a_n is the last term.

n = no of terms

a = first term

d = common difference

Explanation:

Let the a be first term and d be common difference of an AP

As n = 37 (odd )

Middle term will be

Three middle most terms will be

18^th, 19^th and 20^th terms

Given,

a₁₈ + a₁₉ + a₂₀ = 225

a + 17d + a + 18d + a + 19d = 225 [ using a_n = a + ( n - 1 )d]

3a + 54d = 225

3a = 225 - 54d

a = 75 - 18d [ eqn 1]

Also last three terms will be 35^th , 36^th and 37^th terms

Given,
a₃₅ + a₃₆ + a₃₇ = 429

a + 34d + a + 35d + a + 36d = 429

3a + 105d = 429

a + 35d = 143

75 - 18d + 35d = 143 [ using eqn1]

17d = 68

d = 4

using this value in eqn 1

a = 75 - 18( 4 )

a = 3

so AP is a, a + d, a + 2d….

i.e. 3, 7, 11….