Final answer:
To find the first term of a geometric progression (GP) given the third and seventh terms, we can set up a system of equations and solve for the first term and the common ratio. By dividing the equations and solving for the common ratio, we find that the common ratio is 2. Substituting the common ratio into one of the original equations, we can solve for the first term, which is 6.
Step-by-step explanation:
To find the first term of the geometric progression (GP), we can use the formula for the nth term of a GP:
an = a1 * r(n-1)
Given that the third term is 24, we have:
24 = a1 * r(3-1)
Simplifying, we get:
24 = a1 * r2
The seventh term is 4 times the first term:
4 = a1 * r(7-1)
Simplifying, we get:
4 = a1 * r6
We now have a system of two equations with two variables. By dividing the second equation by the first equation, we can eliminate a1:
4/24 = (a1 * r6) / (a1 * r2)
Simplifying, we get:
1/6 = r4
Since r is a common ratio, looking for a value that satisfies the equation and matches the given options, we find r = 2.
Substituting r = 2 into any of the original equations, we can find a1:
24 = a1 * 22
Simplifying, we get:
24 = a1 * 4
Dividing both sides by 4, we find:
a1 = 6
Therefore, the first term of the geometric progression is 6.