To find the first term and the common ratio of a geometric progression, we can use the formula for the nth term of a geometric progression:
An = A * r^(n-1)
where An is the nth term, A is the first term, r is the common ratio, and n is the term number.
Given that the fourth term (n=4) is 72 and the seventh term (n=7) is 576, we can set up two equations:
72 = A * r^(4-1)
576 = A * r^(7-1)
Let's solve these equations step by step:
Equation 1: 72 = A * r^3
Equation 2: 576 = A * r^6
Now, divide equation 2 by equation 1 to eliminate A:
576/72 = (A * r^6)/(A * r^3)
8 = r^6/r^3
8 = r^(6-3)
8 = r^3
Taking the cube root of both sides, we get:
r = 2
Substituting this value of r into equation 1 to find A:
72 = A * 2^3
72 = A * 8
Dividing both sides by 8, we find:
A = 9
Therefore, the first term (A) of the geometric progression is 9, and the common ratio (r) is 2.