# What is the 8th term of the geometric sequence if #a_3 = 108# and #a_5 = 972#?

In a geometric sequence is valid the following rule

So

By the same token

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Let's divide the two terms we have been given, their formulae and their values:

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To find the 8th term of the geometric sequence, we need to first find the common ratio (r) of the sequence using the given terms (a_3) and (a_5).

Given (a_3 = 108) and (a_5 = 972),

We have:

(a_3 = a_1 \times r^2 = 108)

(a_5 = a_1 \times r^4 = 972)

Dividing the second equation by the first equation:

(\frac{a_5}{a_3} = \frac{a_1 \times r^4}{a_1 \times r^2} = r^2 = \frac{972}{108} = 9)

So, (r = \sqrt{9} = 3)

Now that we have the common ratio, we can find the first term ((a_1)) using the third term ((a_3)):

(a_3 = a_1 \times r^2 = 108)

(a_1 = \frac{a_3}{r^2} = \frac{108}{3^2} = 12)

Now, to find the 8th term ((a_8)), we use the formula for the nth term of a geometric sequence:

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

(a_8 = a_1 \times r^{(8-1)})

(a_8 = 12 \times 3^{(8-1)})

(a_8 = 12 \times 3^7)

(a_8 = 12 \times 2187)

(a_8 = 26244)

So, the 8th term of the geometric sequence is 26244.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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