Video Transcript
A geometric series has a first term of three and a common ratio of five. Find the sum of the first six terms.
Well, a geometric series is a series where there is a common ratio between the terms. And we have a general form for a term in a geometric series. And that is that if you have 𝑎 sub 𝑛 is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one, where 𝑛 is the term number, 𝑎 is the first term, and 𝑟 is our common ratio. Okay, great, so, now, we know what a geometric series is, and we know our general form. Let’s see what the question wants us to find out.
Well, the question wants us to find out the sum of the first six terms. And there’s, in fact, a couple of ways that we can do this. And the first method we’re gonna use is to use a formula. And that formula tells us that sum of the first 𝑛 terms is equal to 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 over one minus 𝑟. Well, in our problem, our 𝑎 is gonna be equal to three cause our first term is three. Our 𝑟 is gonna be equal to five. And that’s cause the common ratio is five. And our 𝑛 is gonna be equal to six because we’re looking at the first six terms.
So therefore, if we substitute in the values, what we’re gonna get is that sum of the first six terms is equal to three multiplied by one minus five to the power of six over one minus five. Which is gonna be equal to three multiplied by negative 15624 over negative four, which is gonna be equal to 11718. So therefore, we can say that the sum of the first six terms is 11718. Okay, great, so, we’ve done that using the formula. And that was our first method. But I did say there’d be another method that we could use.
Well, the other method is to find out what the first six terms are and then add them together. Well, we know that the first term 𝑎, or 𝑎 sub one, is equal to three. So, then, if we use the general formula we’ve got to find the second term, we’re gonna find that 𝑎 sub two, so our second term, is equal to three. Because that was 𝑎, our first term, multiplied by five, that’s cause that was our common ratio, to the power of two minus one. And that’s two because the term number is the second term, so it’s two, and then minus one.
So, this means that it’s gonna be three multiplied by five. That’s because two minus one is one. And five to the power of one is just the same as five, which is gonna be equal to 15. So, that’s our second term. So, now, we can move on to the third term. Well, the third is gonna be equal to three multiplied by five to the power of three minus one, which is gonna be equal to three multiplied by five squared. And that’s cause five squared is 25, so three multiplied by 25 is 75.
Then, we do three multiplied by five to the power of four minus one is our fourth term, which is three multiplied by five cubed. Well, five cubed is 125, so three multiplied by 125 is 375. Then, the next two terms are 1875, which is three multiplied by five to the power of four, and 9375, which is three multiplied by five to the power of five. So, then, all we need to do is add these all together.
So, what we have is three add five is eight, add five is 13, add another five, 18, add another five, 23, add another five, 28. So, that’s eight, carry the two. Then, we’ve got one seven is eight. Add seven is 15. Add another seven is 22. Add another seven is 29. Add the two is 31. So, one in the tens column, carry the three. Then, three add eight, which is 11. Add another three is 14. Add the three we had, which is 17. So, we have seven, carry the one. Then, one add nine is 10, add one. Which means we get a final answer of 11718, which is the same as we got with the first method. So, we’ve found the sum of the first six terms is 11718.
