Question

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

Answer 1

$\textsf{Given : The AP consists of 37 terms}$

$\textsf{Given : The Sum of Three Middle Most terms is 225}$

$\textsf{First, Let us find the Middle Most term :}$

$\textsf{As, Total Number of Terms in the AP are 37, We can Divide 37 into 3 parts}$

✿  $\textsf{First Part Contains First 18 terms}$

✿  $\textsf{Second Part Contains only one term which is the Middle Most Term}$

✿  $\textsf{Third Part Contains Last 18 terms}$

$\textsf{From the Above, As the Middlemost term is after first 18 terms : We can say}\\\textsf{that 19th Term is the Middlemost Term}$

$\textsf{Hence, the three most middle terms will be the 19th Term and the}\\ \textsf{immediate terms beside it. They are 18th - 19th - 20th terms}$

$\sf{We\;know\;that,\;In\;an\;AP,\;n^{th}\;term\;is\;given\;by : T_n = a + (n - 1)d }$

$\sf{\implies 18^{th}\;Term\;(T_{18}) = a + (18 - 1)d = a + 17d}$

$\sf{\implies 19^{th}\;Term\;(T_{19}) = a + (19 - 1)d = a + 18d}$

$\sf{\implies 20^{th}\;Term\;(T_{20}) = a + (20 - 1)d = a + 19d}$

$\sf{\implies (a + 17d) + (a + 18d) + (a + 19d) = 225}$

$\sf{\implies 3a + 54d = 225}$

$\sf{\implies a + 18d = 75\;------\;[1]}$

$\textsf{Given : The Sum of Last three terms is 429}$

$\textsf{We can Notice that : The Last three terms are 35th - 36th - 37th terms}$

$\sf{\implies 35^{th}\;Term\;(T_{35}) = a + (35 - 1)d = a + 34d}$

$\sf{\implies 36^{th}\;Term\;(T_{36}) = a + (36 - 1)d = a + 35d}$

$\sf{\implies 37^{th}\;Term\;(T_{37}) = a + (37 - 1)d = a + 36d}$

$\sf{\implies (a + 34d) + (a + 35d) + (a + 36d) = 429}$

$\sf{\implies 3a + 105d = 429}$

$\sf{\implies a + 35d = 143\;-----\;[2]}$

$\textsf{Subtracting Equation [1] from Equation [2], We get :}$

$\sf{\implies (a + 35d) - (a + 18d) = 143 - 75}$

$\sf{\implies a + 35d - a - 18d = 68}$

$\sf{\implies 17d = 68}$

$\sf{\implies d = 4}$

$\textsf{Substituting the value of d in Equation [1], We get :}$

$\sf{\implies a + 18(4) = 75}$

$\sf{\implies a + 72 = 75}$

$\sf{\implies a = 3}$

$\textsf{Hence, The Given Arithmetic Progression is :}$

$\bigstar\;\;\;\textsf{3 , 7 , 11 , 15 . . . . . .139 , 143 , 147}$