The 4th and 7th terms of a G.P. are 16 and 2 respectively.
Find:
a. The common ratio
b. The first term
a. From the formula for the nth term of a G.P., we know that:
𝑎_𝑛 = 𝑎_1 × 𝑟^(𝑛−1)
So we can write:
𝑎_4 = 𝑎_1 × 𝑟^(4−1) = 16
𝑎_7 = 𝑎_1 × 𝑟^(7−1) = 2
Dividing the second equation by the first, we get:
(𝑎_1 × 𝑟^6) / (𝑎_1 × 𝑟^3) = 2/16
Simplifying, we get:
𝑟^3 = 1/4
Taking the cube root of both sides, we get:
𝑟 = 1/2
Therefore, the common ratio is 1/2.
b. We can use the first equation above to solve for 𝑎_1:
𝑎_1 = 16 / 𝑟^3
Substituting the value we found for 𝑟, we get:
𝑎_1 = 16 / (1/8) = 128
Therefore, the first term is 128.
To find the common ratio (r) and the first term (a) of the geometric progression (G.P.), we can use the formula for the nth term of a G.P.:
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 position of the term in the G.P.
Given information:
4th term (A4) = 16
7th term (A7) = 2
To find the common ratio (r):
Step 1: Substitute the given values into the formula for the 4th and 7th terms.
A4 = a * r^(4-1) = 16
A7 = a * r^(7-1) = 2
Step 2: Write the equations for the 4th and 7th terms.
a * r^3 = 16 ----(1)
a * r^6 = 2 ----(2)
Step 3: Divide equation (2) by equation (1) to eliminate 'a'.
(a * r^6)/(a * r^3) = 2/16
r^3 * r^6 = 1/8
r^9 = 1/8
Step 4: Take the cube root of both sides of the equation.
(r^9)^(1/3) = (1/8)^(1/3)
r^3 = 1/2
r = (1/2)^(1/3)
r = 1/2
Therefore, the common ratio (r) is 1/2.
To find the first term (a):
Step 1: Substitute the value of r into equation (1).
a * (1/2)^3 = 16
a * 1/8 = 16
Step 2: Multiply both sides of the equation by 8 to solve for 'a'.
a = 16 * 8
a = 128
Therefore, the first term (a) is 128.
In summary:
a. The common ratio (r) is 1/2.
b. The first term (a) is 128.