Question

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

Answer 1

As it is given that

Fifth term is $19$ i.e $a_5=19$

Also, the difference of the eighth term from the thirteenth term is $20$.

i.e ${13}^{th}term–8^{th}term=20$

Now, We have to determine the AP

We know that the $n^{th}$ term of an AP is given by the formula

$a_n=a+(n–1)d$

where,

$a=$ first term, $a_n$ is the $n^{th}$ term, $d$ is the common difference

As we have $a_5=19$

Using the $n^{th}$ term formula,

$$\begin{gathered} a_n=a+(n–1)d \\ \Rightarrow a+4d=19 \\ \Rightarrow a=19–4d\dots \left(1\right) \end{gathered}$$

Now

$$\begin{gathered} 13thterm–8thterm=20 \\ \Rightarrow a+12d–(a+7d)=20 \\ \Rightarrow 5d=20 \\ \Rightarrow d=4 \end{gathered}$$

On substituting $d=4$ in equation 1,

We obtain,

$$\begin{gathered} \Rightarrow a=19–4\left(4\right) \\ \Rightarrow a=3 \end{gathered}$$

Then, the AP can be expressed as

$3,3+4,3+2\left(4\right),\dots$

Hence, $\mathrm{AP}\mathrm{is}3,7,11,\dots$