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