Math, asked by SharmaShivam, 1 year ago

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.

Answers

Answered by Grimmjow
26

\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}

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