Step 1: General formula of a G.P.

The general term of a G.P. is:

$T_n=a{r}^{n-1}$Tn​=arn−1

Where:

• $a$a = first term

• $r$r = common ratio

• $T_n$Tn​ = nth term.

ep 2: Use given terms

• 2nd term: $ar=6$ar=6 → (1)

• 5th term: $a{r}^4=48$ar4=48 → (2)

Divide (2) by (1):

$\frac{a{r}^4}{ar}=\frac{48}{6}\Rightarrow r^3=8\Rightarrow r=2$arar4​=648​⇒r3=8⇒r=2

Now substitute $r=2$r=2 into equation (1):

$a\left(2\right)=6\Rightarrow a=3$a(2)=6⇒a=3

Step 3: Find the first four terms

Using $a=3$a=3, $r=2$r=2:

• 1st term: $a=3$a=3

• 2nd term: $ar=6$ar=6

• 3rd term: $a{r}^2=12$ar2=12

• 4th term: $a{r}^3=24$ar3=24

Step 4: Sum the first four terms

$S=3+6+12+24=45$S=3+6+12+24=45

✅ Final Answer:

$\boxed{{\displaystyle 45}}$45​