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Determine the AP whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.
Given:
The 5th term of an A.P. is 19 and the difference of the eighth term from the thirteenth term is 20.
To do:
We have to find the AP.
Solution:
Let the first term, common difference and the number of terms of the given A.P. be $a, d$ and $n$ respectively.
We know that,
nth term of an A.P. $a_n=a+(n-1)d$
Therefore,
$a_{5}=a+(5-1)d$
$19=a+4d$
$a=19-4d$.....(i)
$a_{8}=a+(8-1)d$
$=a+7d$....(ii)
$a_{13}=a+(13-1)d$
$=a+12d$....(iii)
According to the question,
$a_{13}-a_8=20$
$a+12d-(a+7d)=20$
$a+12d-a-7d=20$
$5d=20$
$d=4$
$a=19-4(4)$ (From (i))
$a=19-16$
$a=3$
Therefore,
$a_1=a=3, a_2=a+d=3+4=7, a_3=a+2d=3+2(4)=3+8=11$
Hence, the required arithmetic progression is $3,7, 11, .......$
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