Identify the 9th term of the given geometric sequence. [/tex]1/3, −1/9, 1/27, −1/81, HELP ASAP PLEASE!!!

Step-by-step explanation:

Before we find the ninth term, we need to find the common ratio. The common ratio in a geometric sequence is the number that every number after the first one is multiplied by. For example, in the sequence {2, 4, 8, 16, 32, ...}, the common ratio is 2.

Now, for our example, the numbers in the sequence alternate between negative and positive numbers. What does that tell us about the common ratio? Well, the common ratio has to be negative because the negative sign keeps alternating.

So we know the common ratio is negative. But what about the rest of the number? Well, we can see that the ratio has a 1 in the numerator because the numerator is the same throughout the sequence (aside from the alternating negative sign).

Now that we've narrowed down the possibilities to -1/x, let's figure out the denominator of the common ratio. The denominator changes from 3 to 9 to 27 to 81. Do you see a pattern there? Each denominator is 3 times the value of the previous denominator. The denominator of the common ratio, then, must be 3.

Ok. We have our common ratio, -1/3, but the question we still need to find the 9th term in the sequence. Well, we can write out the pattern until we get to the term we want. But what if you were trying to find the one hundredth or the one thousandth term? That would take a very long time to write out. Instead, we can take the first number in the sequence (1/3) and multiply it by the our common factor to the power of the remaining terms. As a formula, this looks like: , where a is your first term, f is your common factor and n is the number of the term you are trying to find.

So, finally, we can solve the problem.

1/3 * (-1/3)⁸ = a₉

1/3 * 1/6561 = 1/19683

So the value of the ninth term of the given geometric sequence is 1/19683.

I hope you learned something and it wasn't too difficult to understand.

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step-by-step explanation:

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