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Q.
If the sum of the first 2n terms of the A.P. 2, 5, 8, ........... is equal to the sum of the first n terms of the A.P.
57, 59, 61, ........, then n equals
Sequences and Series
Solution:
$ 2,5,8, \ldots \ldots . . . . . A . P . $
$S _{2 n }= n [4+(2 n -1) 3] $
$S _{2 n }= n [6 n +1] $
$57,59,61, \ldots \ldots . . . .$
$S _{ n }=\frac{ n }{2}[2 \times 57+( n -1) \times 2]$
$S _{ n }= n [57+ n -1]= n [ n +56] $
$S _{ n }= S _{2 n }$
$n ( n +56)= n (6 n +1) $
$n \neq 0 \therefore n +56=6 n +1$
$5 n =55 $
$\therefore n =11 $