Use the formula for the sum of the first n terms of a geometri...

Question

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.

$\sum_{i = 1}^{6} 5^{i}$

Accepted Answer

Welcome back. I'm so glad you're here. We've got to find the sum of the first N terms is indicated. And we know from previous lessons that in sigma notation, the N number here is right up on top of our sigma notation. So we know that N equals nine. Great. Got that part done. And now to find the sum of those terms, let's make sure that we're dealing with a geometric series because it looks like it could be, it looks like there's multiplication going on. And if we are dealing with a geometric series, we have got a formula we can use. So let's figure out what this would be if we were doing it longhand, that would be taking this, I value the first I value is going to be one and plugging that up here in for I. So the first terms value. Our a sub one would be eight to the first and are a sub two would be plugging in a two up here for the I. Value and are a sub three. While that would be plugging a three up here and that's going to keep going all the way up to our ninth value. So eight to the ninth and wo eight to the ninth is going to get big. So is this geometric? Well, we know that by taking a look at the ratio to find the ratio, we take two subsequent terms and we divide the second term of the two that we've chosen this time. I'll choose up to buy the one immediately preceding it. So that would be a sub one, A sub two. We've got eight squared divided by a sub one is eight to the first. In other words we've got 64 divided by eight and our our value here is eight. And now let's test to others to see if they're the same. I'll use a sub three divided by the term immediately preceding it. A sub two. So a sub three is going to be eight cubed divided by a sub two. That was eight squared. Or in other words we've got divided by 64 yes, that is eight. Yes. This is a geometric series which means we can use our formula. We recall from previous lessons that this formula for the sum of the end terms in a geometric series is going to be a sub one value of the first term times one minus R. R ratio raised to the end. That number Divided by one -R. And now we can identify which terms we figured out. Well, a sub one that is eight to the first or eight. Our ratio is eight And RN is nine. Let me change colors here so that these two aren't the same. A sub one is eight. So we'll fill those in a sub one is eight times one minus R. Is also eight raised to the N. Which is nine. And we're going to divide all that by one minus again are Our which is eight. Okay, eight to the ninth. These numbers are going to get big so we can plug this into a calculator And that's going to give us the answer of 153,391, obviously, wow. Thank goodness for calculators that matches with answer choice B. Well done, we'll catch you on the next one.

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.
$\sum_{i = 1}^{6} 5^{i}$

Key Concepts

Geometric Sequence

Sum of the First n Terms of a Geometric Sequence

Sigma Notation

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