Question

Question 5
Determine the AP whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.

Answer 1

Let the first term of an AP be "a" and common difference be "d".

$Given,a_5=19$
$\Rightarrow a_5=a+(5-1)d=19[\because a_n=a+(n-1\left)d\right]$

$and{a}_{13}-a_8=20$
$\Rightarrow [a+(13-1\left)d\right]-[a+(8-1\left)d\right]=20[\because a_n=a+(n-1\left)d\right]$
$\Rightarrow a+4d=\mathrm{19...}\left(i\right)$
$\Rightarrow a+12d-a-7d=20$
$\Rightarrow 5d=20$
$\therefore d=4$
Substituting d = 4 in eq. (i), we get;
a + 4(4) = 19
a + 16 = 19
a = 19 - 16 = 3
So, the required AP is : a, a+d, a+2d, a+3d, ...
= 3, 3+4, 3+2(4), 3 +3 (4), …
= 3, 7, 11, 15, ...