Theory of Intelligence
11TH•Mathematics•Sequences And Series•Question 13

Overview

Find the 8th term from the end of the A.P. $8, 11, 14, \ldots, 185$.

A.P.: ( 8, 11, 14, \ldots, 185 )

( a = 8 ), ( d = 3 ), last term ( l = 185 )

Number of terms: [ 185 = 8 + (n-1)3 \implies (n-1)3 = 177 \implies n-1 = 59 \implies n = 60 ]

8th term from the end is the ( (60 - 8 + 1) = 53 )rd term from the beginning.

[ a_{53} = 8 + 52\cdot 3 = 8 + 156 = 164 ]

Alternatively, the sequence from the end is ( 185, 182, \ldots ) with common difference ( -3 ).

8th term: ( 185 + 7\cdot(-3) = 185 - 21 = 164 )

Answer: 164

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