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. 8 Σ (i = 1) 3^i

Sum of the first 8 terms of the geometric sequence 3^i from i=1 to 8.

Accepted Answer

Everyone in this problem we are asked to evaluate the specified sum for the given geometric sequence. And the some were given is the sum I equals 1-10 of two to the exponents. I okay. Now Were given the sum of a geometric sequence. Okay. And then the general form of the sum of a geometric sequence is going to be the sum from I equals zero two, N -1 of a R. To the exponent I. Okay. And what you'll notice is that here we started I equals zero. But in the somewhere given we started I equals one. Okay. So we want to write the somewhere given to also start at I equals zero. So that we can compare these two to get the values for A R and N. That we need for our some. So we have the some I was 1 - to to the eye. Okay well we can pull out whoops and that's not two times I that's two to the exponent I. Sorry We can pull out a factor of two And we can write this as two Times 2 to the exponent. I oops. And that shouldn't be too to the exponent I when we pull out a factor of two we get two to the X one I -1. Okay so two times two to the I minus one. The same base. So the exponents add and we're just left with two. I like we had originally Now that we have this I -1. We can shift or index. Okay and so we can say let's use K just to indicate that we've shifted the index and we're using a different index from zero. Okay, we shift down by one for both of them and then we have two times two to the K. Because we've shifted. And you can double check on the left hand side we start at I equals one when I is one. We have two times two to the 1 -1. That's gonna be two to the zero, which is just gonna be one. And so we get to when K is zero we get two times two to the zero. So we just get to as well and you can do the same thing for the other terms to see that these are in fact the same. Okay, we've just shifted that index. So now that we have it written like this, we can compare it to the general form to see that A. Is equal to two. R. is equal to two and we know that N -1 is equal to nine or in other words N is equal to 10. Okay. Alright, so recall that the sum of a geometric sequence OK. S is given by a times one minus R. To the end over one minus R. In this case a is two. So we get two times 1 -2 to the exponent 10 Divided by -2. Let's give ourselves a bit more room to work here. We get two. Okay, on the bottom we're gonna get negative one, multiplied by negative one. We can switch the signs up here so we get two times two to the exponent 10 minus one. Which is gonna give us 2,046. Okay? And so the some of the geometric sequence we were given is 2,046. That is going to be answered. B. Thanks everyone for watching. See you in the next video.

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 8 Σ (i = 1) 3^i

Key Concepts

Geometric Sequence

Sum of a Geometric Series

Sigma Notation

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