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The sum of the 5th and the 9th terms of an AP is 30. If its 25th term is three times its 8th term, find the AP.
Given:
The sum of the 5th and the 9th terms of an AP is 30.
25th term is three times its 8th term.
To do:
We have to find the AP.
Solution:
Let the first term of the AP be $a$ and the common difference be $d$.
This implies,
nth term $=a+(n-1)d$
5th term$=a+(5-1)d=a+4d$
9th term$=a+(9-1)d=a+8d$
$a+4d+a+8d=30$ (Given)
$2a+12d=30$
$2(a+6d)=2(15)$
$a+6d=15$
$a=15-6d$....(i)
8th term$=a+(8-1)d=a+7d$
25th term$=a+(25-1)d=a+24d$
$a+24d=3(a+7d)$ (Given)
$a+24d=3a+21d$
$3a-a=24d-21d$
$2a=3d$
$2(15-6d)=3d$ (From (i))
$30-12d=3d$
$30=12d+3d$
$15d=30$
$d=\frac{30}{15}$
$d=2$
Therefore,
$a=15-6(2)=15-12=3$
$a+d=3+2=5$
$a+2d=3+2(2)=3+4=7$
The required AP is $3, 5, 7, .....$.