Final answer:
To find the first term of a geometric progression (GP) when the fifth term is 48 and the seventh term is 8 times the fourth term, we can solve for the common ratio and substitute it into the formula to find the value of the first term. The first term of the GP is 3.
Step-by-step explanation:
To find the first term of a geometric progression (GP) when the fifth term is 48, we need to determine the common ratio first. Given that the seventh term is 8 times the fourth term, we can write the equation:
7th term = 8 * 4th term
Using the formula for the nth term of a GP (where 'a' is the first term and 'r' is the common ratio), we can write:
a * r^6 = 8 * (a * r^3)
Canceling the 'a' on both sides, we get:
r^6 = 8 * r^3
Dividing both sides by r^3, we get:
r^3 = 8
Taking the cube root of both sides, we get:
r = 2
Now, to find the first term, we can use the formula for the fifth term:
5th term = a * r^4
Substituting the given value and the calculated common ratio:
48 = a * 2^4
Simplifying the equation:
48 = 16a
Dividing both sides by 16, we get:
a = 3
Therefore, the first term of the GP is 3.