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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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