In Exercises 31–36, find the indicated sum. Use the formula for t...

Question

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 6 Σ (i = 1) (1/2)^(i + 1)

Accepted Answer

Hey everyone in this problem we are asked to evaluate the specified sum for the given geometric sequence. Okay. And were given the sum from 0 to 7 of one third. All to the exponent k plus one. Okay. All right. So let's recall the general format of a sum of a geometric sequence. If we have a sum of a geometric sequence, we typically write it as a sum K equals zero two, N - of a. Are the common ratio to the exponent k. Okay. And so when we compare this general format to the equation were given or the some were given, you'll see that in the general equation we have to the exponent K and in the one we're given we have to the exponent K plus one. Okay, so how can we kind of manipulate what we're given? So that it looks like this general formula. Okay, well if we have the sum of one third to the K plus one. When we have an exponents we have two things. Same base. We can add the exponents when we multiply them. Okay, so what do I mean what I mean is we can have one third. Okay, pull out a factor of one third and then we will have one third to the exponent K left. Okay, these have the same base. So when you multiply them you add the exponents so you multiply them, you get one third to the exponent K plus one which is exactly what we have. Okay, so it's like we just took out a factor of a third. Now this looks like our general format. Okay. Where we have a is equal to a third. We have r. is equal to a third. Okay. And we have N -1 is equal to seven Or N is equal to eight. Okay. Alright. Now we're asked to evaluate this some. Now let's recall that the some s for a geometric sequence can be written as a times one minus R. To the exponent end Divided by one -R. Okay well we have our a we have our our we have our end. So let's go ahead and we're going to have one third Times 1 -1 3rd to the exponent eight Divided by 1 -1 3rd. All right now at this point you want to double check depending on your instructor and if it's a multiple or if it's a multiple choice question you can look and see you want to figure out do I want to answer this infractions or do they want decimals? Okay, so we look at the possible answer choices here we see we have fractions. So we wanna continue with fractions. In some cases you'll want to be using decimals. Okay so always just check at this stage before you keep going. Alright so let's move down we want to stick with fractions give ourselves some more room to work out all of these terms. So this is going to give us 1/ times one minus 1/3 to the exponent eight divided by 2/3. This is gonna give us one one third. The exponents eight. Okay a third divided by 2/3. It's gonna be over too. So we get one minus working out this exponent, we get three to the eight. So we get one over 6561. Okay. All divided by two. Okay finding a common denominator in the numerator We are going to get 6561 -1, Divided by 6561. All of this divided by two, this is going to give us 6,568 divided by two And 6561. Okay. And you can see in the top we have an even number and we have this factor of two in the bottom. And so we can divide to get our final answer. 3280 divided by 6561. Okay. And so are some s of the geometric sequence is going to be 3280 divided by 6561 And if we go back up to our answer choices, we will see that this is going to correspond with answer. A thanks everyone for watching. I hope this video helped see you in the next one

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 6 Σ (i = 1) (1/2)^(i + 1)

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