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 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...

Accepted Answer

everyone in this problem. We are asked to evaluate the some of the 1st 10 terms for the geometric sequence given below were given 36 6, 1/6 and 1/36. Okay, so let's recall the general sum of a geometric sequence. Okay, we get the sum K equal 0 to N - of a times R to the exponents. Okay now the first thing we want to find is our okay our is this common ratio? And so that's gonna be the ratio between a term and it's proceeding term. Okay, so between two consecutive terms. So in order to find our what we do is we take a term and we divide it by the proceeding term. Okay, so if we start with the first two terms we have six. We divided by the preceding term 36 we get an R. Value of 1/6, you can do this with the other terms as well. Just to be sure that this is a geometric sequence and that the r is consistent. Okay, so if we do the next two terms, we have one divided by the preceding terms six which is 1/6. And so are is that ratio between the two terms? That's what you have to multiply a term by to get the next term. Okay, if we go to the next term we have 1/ divided by one. That's also gonna be 16 In the last two terms we have won over 36 divided by 1/ and this is also going to give us 16. Okay. And so the r value, it is 1/6. It's consistent. That is a geometric sequence that we've been given now A A is going to be the first term. How do we know this? Well the first term is when K is zero. Okay, so A are to the zero? Well are to the zero is just gonna be one so we get a Okay. And we know that the first term is 36. Okay, so we have R is equal to 16. A is 36 so are some looks like this, K equals zero. Now what do we go to? And is the number of terms we're told to do the 1st 10 terms? So we get 10 minus one. Okay, at 36 times are 16 to the exponent in this minus one. Accounts for the fact that we're starting at zero. Okay, so we have 10 terms but we start at zero. So we're going to go 0 to 9 instead of 1 to 10. Alright, now we can find the sum. Okay. And let's recall that the some s have a geometric sequence is given by a Times one -R to the end Divided by one -R. Well in our case this is going to be 36 times one minus 1/6 To the exponent. 10 Divided by 1 - 6th. All right now Well, at this stage you want to check either check with your instructor on what they expect or check if this is a multiple choice problem at what the solutions look like. Okay. And are possible answers are fractions. So we need to work with fractions and complete this infractions instead of converting to a decimal. Ok, so check at this stage before you continue on whether you should be working with fractions or whether you should go ahead and convert to a decimal. Ok. Alright. We're working with fractions. So what do we get? We have 36. Okay. This is gonna be 1 -1 over this big number,  aN:aN:000NaN 60466176. Okay, that's 6 to the experiment 10. All of that divided by one minus a six which is 56. Alright, simplifying. We get 36. Okay, times six divided by five. Okay, so this is the 36 divided by 56. Is 36 times six divided by five. Okay. And then using a common denominator for here we're gonna get 60,466, minus one. All divided by that same big number. All right, We can simplify that a little bit. 36 Times six divided by five Times divided by The same 60,466,176. Okay. Now We want to simplify this as much as we can and what you'll notice is we have this big term in the top that ends in a five. We have a factor of five in the bottom. So we can divide by five. Okay, let me move down so we have some more room to work. So if we divide by that factor of five, we're gonna be left with 36 times six times 60,466, five. And sorry I forgot the division part there writing the same line again. Sorry, when we divide by five, we're gonna have 12,093,235. There we go. That's our division by five, 60466176 in the Denominator. And we can actually divide again by these factors of 36 and six. We're gonna end up with 12 million 93, 235 in the numerator, 279936 in the denominator. And so this is the sum s of the geometric sequence that we were given. And if we go back up to those answer choices, we see that this is going to correspond with answer C. Okay, 12,093,235. Over Sorry, 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 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...