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mylen [45]
3 years ago
11

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
Mathematics
1 answer:
IgorLugansk [536]3 years ago
6 0

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

<h3>How to determine the sequence?</h3>

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

a_n = a_1*(r)^{n-1}

Here we do know:

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

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:

\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

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:

16 = a_1*(1.084)^{22}\\\\a_1 = \frac{16}{1.084^{22}}  = 2.713

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

a_n = 2.713*(1.084)^{n-1}

Then the 43th term is:

a_{43} = 2.713*(1.084)^{43-1} = 80.3

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

brainly.com/question/7882626

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