The 4th and 7th terms of a series in G.P are 48 and 384 respectfully. Find the first term and the common ratio

Q. The 4th and 7th terms of a series in G.P are 48 and 384 respectfully. Find the first term and the common ratio.
Solution:
Let us assume that first term=a and common ratio=r.
Given,4th term=48.
⇒ar⁽⁴⁻¹⁾⁼48
⇒ar³=48...........................................(i)
and, 7th term=384
⇒ar⁽⁷⁻¹⁾=384
⇒ar⁶=384.......................................(ii)
(now we will divide equation (ii)by (i) )
(ii)÷(i)⇒(ar⁶)÷(ar³)=384÷48
⇒r³=8
(by the formula of indices we get it.

here first of all, a is cancelled out as it is present in both numerator and denominator.then the remaining value of power of r is following by this formula. That means 6-3=3)
(Now in the LHS the value is in the power of r and the power is 3, so we will try to express the RHS value in the power of 3. We will concentrate to express the value in the power of 3. We know that 2³=8. So we will write the RHS value in the form 2³=8)

⇒r³=2³
⇒r=2
(because again from the indices formula, 

, If you compare the value of the above line with this formula then you can see that n=3 in the line,a=r and b=2. So by using this formula we can write r=2)
Now by substituting the value r=2 to the equation (i), we get,
a2³=48
⇒a*8=48
⇒a=48÷8
⇒a=6
Therefore, first term=6, common ratio=2.