# How do you find a formula of the nth term if the 4th term in the geometric sequence is -192 and the 9th term is 196608?

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The general term of a geometric sequence can be described by the formula:

Given:

We find:

This yields one possible Real value for the common ratio:

Then:

So if our sequence is of Real numbers, then the only solution is:

Then the possible values for the common ratio are:

with corresponding initial terms:

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To find the formula for the nth term of a geometric sequence, we need to first find the common ratio (r) of the sequence. We can do this by dividing any term by the previous term. Then, we can use the formula for the nth term of a geometric sequence, which is:

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

Given that the 4th term is -192 and the 9th term is 196608, we can find the common ratio (r) by dividing the 9th term by the 4th term:

[ r = \frac{196608}{-192} ]

Now that we have the common ratio, we can use it to find the first term (a₁) of the sequence. We can use either the 4th or 9th term along with the common ratio to find ( a_1 ).

Let's use the 4th term:

[ -192 = a_1 \times r^{(4-1)} ] [ -192 = a_1 \times r^3 ]

Now, we can solve for ( a_1 ):

[ a_1 = \frac{-192}{r^3} ]

Now that we have ( a_1 ) and ( r ), we can write the formula for the nth term of the geometric sequence:

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

Substituting the values we found:

[ a_n = \left(\frac{-192}{r^3}\right) \times r^{(n-1)} ]

Simplify:

[ a_n = -192 \times r^{(n-4)} ]

Therefore, the formula for the nth term of the geometric sequence is:

[ a_n = -192 \times \left(\frac{196608}{-192}\right)^{(n-4)} ]

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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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