The 23rd term in a certain geometric sequence is 16 and the 28th term in the sequence is 24. What is the 43rd term?
To find the 43rd term in the geometric sequence, we first need to find the common ratio of the sequence.
In a geometric sequence, each term is found by multiplying the previous term by a constant value called the common ratio. Let's denote the first term of the sequence as a₁, and the common ratio as r.
We are given that the 23rd term is 16, which we can represent as a₃ = 16, and the 28th term is 24, which we can represent as a₃ = 24.
To find the common ratio (r), we can use the formula for the nth term of a geometric sequence:
aₙ = a₁ * r^(n-1)
Using the value of a₃ = 16, we can write the following equation:
16 = a₁ * r^(3-1)
Simplifying:
16 = a₁ * r^2
Likewise, using the value of a₃ = 24, we can write the following equation:
24 = a₁ * r^(8-1)
24 = a₁ * r^7
Now we have a system of equations:
16 = a₁ * r^2 (Equation 1)
24 = a₁ * r^7 (Equation 2)
We can solve this system of equations to find the values of a₁ and r.
Divide Equation 2 by Equation 1:
24/16 = (a₁ * r^7) / (a₁ * r^2)
3/2 = r^5
To solve for r, we take the fifth root of both sides:
(r^5)^(1/5) = (3/2)^(1/5)
r = (3/2)^(1/5)
Now that we have the common ratio (r), we can find the first term (a₁) by substituting the value of r into either Equation 1 or 2. Let's use Equation 1:
16 = a₁ * r^2
Substituting r = (3/2)^(1/5):
16 = a₁ * ((3/2)^(1/5))^2
Simplifying:
16 = a₁ * (3/2)^(2/5)
To find the value of a₁, we divide both sides by ((3/2)^(2/5)):
a₁ = 16 / ((3/2)^(2/5))
Now that we have found the first term (a₁) and the common ratio (r), we can find the 43rd term (a₄₃) using the formula for the nth term:
a₄₃ = a₁ * r^(43-1)
Evaluating this expression will give us the 43rd term in the geometric sequence.
d=(T28-T23)/(28-23) = (24-16)/5 = 8/5
T43 = T28+15d = 24 + 15(8/5) = 24+24 = 48