A Gp has a first term of a common ratio of 6r and its 3rd term is 3456 evaluate a and r
The formula for the nth term of a geometric progression (GP) is given by:
an = a * r^(n-1)
Given that the first term (a) has a common ratio (r) of 6r, we can rewrite the formula as:
a(n) = (6r) * r^(n-1)
We are given that the 3rd term (a3) is 3456. Substituting the values into the formula, we have:
3456 = (6r) * r^(3-1)
3456 = (6r) * r^2
3456 = 6r^3
To solve for r, we divide both sides of the equation by 6:
r^3 = 3456/6
r^3 = 576
Taking the cube root of both sides gives us:
r = ∛(576)
r ≈ 8
Now that we have found the value of r, we can substitute it back into the equation to solve for a:
a(1) = 6r
a(1) = 6 * 8
a(1) = 48
Therefore, the first term (a) is 48 and the common ratio (r) is 8.
To find the values of a and r in the geometric progression (GP), we can use the formula for the nth term of a GP:
an = a * r^(n-1)
Given that the first term (a1) is a common ratio of 6r, we can rewrite it as:
a1 = 6r
The third term (a3) is given as 3456. Using the formula, we can calculate it as:
a3 = a * r^(3-1)
3456 = 6r * r^2
3456 = 6r^3
Now, we can solve this equation to find the values of a and r.
Divide both sides of the equation by 6:
r^3 = 576
Take the cube root of both sides:
r = ∛576
Using a calculator, we find:
r ≈ 8
Now, substitute the value of r into the equation a = 6r:
a = 6 * 8
a = 48
Therefore, the values of a and r are a = 48 and r = 8 in the given geometric progression.