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svp [43]
3 years ago
15

The 5th term in a geometric sequence is 40. The 7th term is 10. What is (are) the possible value(s) of the 4th term?

Mathematics
1 answer:
Lena [83]3 years ago
3 0

Answer:

possible values of 4th term is 80 & - 80

Step-by-step explanation:

The general term of a geometric series is given by

a(n)=ar^{n-1}

Where a(n) is the nth term, r is the common ratio (a term divided by the term before it) and n is the number of term

  • Given, 5th term is 40, we can write:

ar^{5-1}=40\\ar^4=40

  • Given, 7th term is 10, we can write:

ar^{7-1}=10\\ar^6=10

We can solve for a in the first equation as:

ar^4=40\\a=\frac{40}{r^4}

<em>Now we can plug this into a of the 2nd equation:</em>

<em>ar^6=10\\(\frac{40}{r^4})r^6=10\\40r^2=10\\r^2=\frac{10}{40}\\r^2=\frac{1}{4}\\r=+-\sqrt{\frac{1}{4}} \\r=\frac{1}{2},-\frac{1}{2}</em>

<em />

<em>Let's solve for a:</em>

<em>a=\frac{40}{r^4}\\a=\frac{40}{(\frac{1}{2})^4}\\a=640</em>

<em />

Now, using the general formula of a term, we know that 4th term is:

4th term = ar^3

<u>Plugging in a = 640 and r = 1/2 and -1/2 respectively, we get 2 possible values of 4th term as:</u>

ar^3\\1.(640)(\frac{1}{2})^3=80\\2.(640)(-\frac{1}{2})^3=-80

possible values of 4th term is 80 & - 80

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