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Find the sum of the first six terms of the geometric series a half plus a quarter plus an eighth plus a 16th and so on.
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So the first thing to do in this type of questions is that actually letβs think what is a geometric series?
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Well, a geometric series is actually a series with a common ratio between successive terms.
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So that actually means that if I divide a term by the previous term, itβs going to give us the same value each time.
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Okay, great, so now we know what it is.
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But how do we actually work out the sum of the first six terms?
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Well, to work out the sum of any number of terms, we actually have a formula.
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And the formula is that the sum of the number of terms β so π π β is equal to π and then one minus π to the power of π divided by one minus π.
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And this is when π is equal to the first term and π is equal to the common ratio.
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Okay, so we actually have this formula.
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Letβs use it to find the sum of our first six terms of the geometric series.
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Well, the first thing we need to do is actually find out what π and π are.
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Well, π is going to be a half because itβs our first term.
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So weβve got π.
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Now what π going to be?
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Well, as we said, π is gonna be our common ratio.
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So what we need to do actually to find out our common ratio is actually find a term and then divide it by the previous term.
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And that will give us our common ratio.
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So Iβm gonna take our second and first terms.
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So weβre gonna say that π is equal to a quarter which is our second term divided by our first term which is a half.
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And then, we can say this is equal to a quarter multiplied by two over one because remember if weβre dividing by a fraction, weβre actually find the reciprocal and multiply.
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So therefore, we can say that π is gonna be equal to two over four which is equal to a half.
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Okay, great, weβve now found our common ratio.
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Okay, now, we found our π and π.
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There is one more variable that we need to know before we substitute into our formula.
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And thatβs π.
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Well, π we said was actually the number of terms.
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And if we look back at the question, we can see that- ok, well, the number of terms is gonna be six because we want to find the sum of the first six terms.
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So now weβve got all the variables we need to actually substitute back into the formula.
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So letβs do that and find the sum of our first six terms.
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So weβre gonna get the sum of the first six terms is equal to a half multiplied by one minus a half to the power of six divided by one minus a half which is gonna be equal to a half multiplied by one minus one over 64 and then divided by a half.
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Well, therefore, we can actually divide top and bottom β so the numerator and the denominator by a half.
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So therefore, we can say that the sum of the first six terms is gonna be equal to 63 over 64.
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And we got that because if you have one minus one over 64, that will be like 64 over 64 minus one over 64, which gives us 63 over 64.