Use the formula for the sum of the first n terms of a geometric s...

Question

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30.

Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...

Accepted Answer

hey everyone in this problem we are asked to evaluate the some of the first eight terms for the geometric sequence given below the sequence we're given is five negative 7.5. 11.25 and negative 16.875. Okay, so let's recall the general form for the sum of a geometric sequence. It's going to be the sum K from zero to n minus one. Where n is the number of terms any time A times R to the experiment K. All right. So let's start by finding our our is going to be our common ratio. Okay. And what that means, R O is going to be the number that we need to multiply a term by in order to get to the next room in an arithmetic sequence. The difference between the terms is constant and a geometric sequence. The ratio between the terms is constant. Okay, we're going to find that common ratio are and in order to do that, we're going to divide consecutive terms. So we're gonna take the second term, -7.5. And we're going to divide it by the first term. Okay. And this is going to give us an r value of -3/2. Alright, let's just double check that that is consistent and this really is a geometric sequence. Okay, so we can do the same for the next two terms. So we have 11.25 Divided by -7.5. And this is going to give us again negative three halves And we can do it one more time with the last terms. 16.875 divided by 11.25 and again negative three have so as we expected we had the common ratio. Are it's consistent as we would have wanted for a geometric sequence. Okay, so we found our what about a well A is just going to be the first term. Okay, so A is going to be five so A is equal to five. R is equal to -3/2. And and while we're told we want the first eight terms and it's going to be eight. So are some looks like this? The sum from K equals 02, -1 which is seven five times negative 3/2 to the exponent. Okay, right now we're asked to find the sum. Let's recall that the sum of a geometric sequence can be given by S is equal to a times one minus r to the end over one minus. Okay, filling this in with the information, we have this is equal to five times -3/2 to the exponent. eight divided by one minus negative three halves. And at this point you want to double check if this is a multiple choice question. You want to double check. Are they using decimals in the answer or are they using fractions? And if it's not a multiple choice question you should know what your instructor prefers, whether they prefer you to use decimals or whether they prefer to keep this infractions. Okay. Um, in this case the possible answers given are in decimal form. So let's go ahead And plug this into our calculator and get the decimal value. And this is going to be negative . 8125. And that is going to give us that is going to be sorry are some of the geometric sequence we were given that is going to correspond with answer. D. Thanks everyone for watching. I hope this video helped see you in the next one.

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Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...