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The $23^\text{rd}$ term in a certain geometric sequence is 16 and the $28^\text{th}$ term in the sequence is 24. What is the $43^\text{rd}$ term

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facundo3141592

By solving a system of equations, we will see that the 43th term of the sequence is 80.3

How to determine the sequence?

We know that the n-th term of a sequence is given by:

[tex]a_n = a_1*(r)^{n-1}[/tex]

Here we do know:

[tex]a_{23} = 16 = a_1*(r)^{22}\\\\a_{28} = 24 = a_1*(r)^{27}[/tex]

Basically, we have a system of equations that we can use to find the value of r and the first term of the sequence. If we take the quotient of the two above equations we get:

[tex]\frac{24}{16} = \frac{a_1*(r)^{27}}{a_1*(r)^{22}} \\\\1.5 = r^{27 - 22} = r^5\\\\\sqrt[5]{1.5} = r = 1.084[/tex]

Now we know the value of r, we can use it to find the value of the first term, I will use the first equation:

[tex]16 = a_1*(1.084)^{22}\\\\a_1 = \frac{16}{1.084^{22}} = 2.713[/tex]

Now we know that the n-th term of our sequence is given by:

[tex]a_n = 2.713*(1.084)^{n-1}[/tex]

Then the 43th term is:

[tex]a_{43} = 2.713*(1.084)^{43-1} = 80.3[/tex]

If you want to learn more about sequences, you can read:

https://brainly.com/question/7882626

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