The missing terms in A.P.: ______, $13$, ______, $3$ are

Let the four terms of the Arithmetic Progression be $a_1,a_2,a_3,$ and $a_4$.
We are given:

* Second term $a_2=13$
* Fourth term $a_4=3$

Using the AP general nth-term expression $a_n=a_1+(n-1)d$:

1. $a_1+d=13$
2. $a_1+3d=3$

Subtracting equation (1) from equation (2) to solve for common difference ($d$):

$$(a_1+3d)-(a_1+d)=3-13$$

$$2d=-10⟹d=-5$$

Substitute $d=-5$ back into equation (1):

$$a_1+(-5)=13⟹a_1=18$$

Now, solve for the third missing term $a_3$:

$$a_3=a_2+d=13+(-5)=8$$

The missing terms $a_1$ and $a_3$ are $18$ and $8$ respectively.