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 12 terms of the geometric sequence: 2, 6, 18, 54, ...

Accepted Answer

hey everyone in this problem we are asked to evaluate the some of the 1st 11 terms for the geometric sequence given below were given four, negative 10, 25 negative 125 over two. Okay, now let's think about what the sum of a geometric sequence looks like in general. Okay. We have the sum K Equal 0 to N -1. Where N is the number of terms Okay. Of a R to the experiment. Okay. Now r is going to be the common ratio. Okay, but what that means is in order to go from one term to the next term, you multiply by our Okay, in an arithmetic sequence, the difference between the terms is the same. It's constant. In a geometric sequence, the ratio between the terms is constant and that constant ratio is our. Okay, so let's find that ratio are okay. And in order to do that we are going to take the ratio of consecutive terms. So let's take the second term divided by the first term. We get negative 10 divided by four which gives us an R value of negative five Halfs. Ok, now we're told this is a geometric sequence. So that should be our art. It should be consistent for all of those terms. Let's just double check to make sure that this is indeed a geometric sequence. Before we continue. Okay, so we do negative 10 divided by four. Let's move on to the next two terms. So now we have the next term 25 divided by the preceding term negative 10 which again gives us an R. Value of negative five paths. If we do it for the last terms negative 125 divided by two Over 25. We again get negative five paths. So as we expected we have the same r. This is a geometric sequence. Okay. With r equals negative by paths. Now what about n? Okay. And well we're told we want to look at the 1st 11 terms, so N is going to be and A is the first term. Okay, well the first term were given is four, so A is going to be four. And are we just found to be negative five halves? Okay so we have all the values we need and we can now write are some some K equal 0 to N -1. So 11 -1 which is going to be 10 A which is four times are negative 5/2 to the exponent K. Alright, so we've written are some how can we evaluate it to evaluate the sum of a geometric series or a geometric sequence? Sorry, let's recall that we have S is equal to a one minus R. To the exponent. End over one minus R. Okay. And s is that some that we're trying to evaluate? Alright, substituting in the numbers we have we get a just four times one minus R. Negative five paths to the exponent N which is 11 divided by one minus R. Which is negative five pass. Okay, Now when we get to this point we want to take a look at the possible answers in this case there are infractions Okay So we want to keep our answer infractions instead of going into decimals um and you can check with your instructor on whether they prefer you stay in with fractions or whether you convert into decimals. Okay? But just double check at this point before you keep going. So we want to work with fractions. So what are we gonna have? Well we're gonna have four. Okay. Times the numerator we have one. Okay. And then we're gonna have plus because we're gonna have a negative to an odd exponents so that's gonna give us a negative negative Negative gives us positive. So one plus this is gonna be 48 million Divided by 2048. Okay so the numerator is negative five to the 11 or five to the 11 and the denominator is two to the 11 and this is all going to be divided by one minus negative five halves. One plus five halves which is seven halves. All right, let's give ourselves some more room to work here. What is this going to give us? Okay, well four divided by seven halves. That's going to be 8/7. Okay let's do a common denominator In order to simplify the fraction in the numerator. So we get 2,048 plus 48,828,125. All divided by Simplifying the numerator, we get 8/7 times 48 million 830, 173 divided by 2048. You'll notice that 2,048 can be divided by eight. Okay? So if we go ahead and do that, we get 48,830,173 in the numerator and the denominator we're left with seven times 2048 divided by eight, which is 256. Okay. And you can see we're just working bit by bit to try to reduce this fraction as much as we can. Okay. And it turns out that we can also divide this numerator by a factor of seven, which will leave us with s equals sorry, six million divided by 256. Okay. And so this is the sum of the geometric sequence we were given. If we go back up to our possible answer choices, we will see That we have answer a the sum is 6,975,739 divided by 256. Thanks everyone for watching. I hope this video helped see you in the next one.

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 12 terms of the geometric sequence: 2, 6, 18, 54, ...

Watch next