Video: Finding the Sum of ๐‘› Terms in a Given Geometric Series

Chris Oโ€™Reilly

Find the sum of the first 6 terms of the geometric series (1/2) + (1/4) +(1/8) + (1/16) + ... .

03:12

Video Transcript

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.

So the first thing to do in this type of questions is that actually letโ€™s think what is a geometric series? Well, a geometric series is actually a series with a common ratio between successive terms. So that actually means that if I divide a term by the previous term, itโ€™s going to give us the same value each time.

Okay, great, so now we know what it is. But how do we actually work out the sum of the first six terms? Well, to work out the sum of any number of terms, we actually have a formula. And the formula is that the sum of the number of terms โ€” so ๐‘† ๐‘› โ€” is equal to ๐‘Ž and then one minus ๐‘Ÿ to the power of ๐‘› divided by one minus ๐‘Ÿ. And this is when ๐‘Ž is equal to the first term and ๐‘Ÿ is equal to the common ratio.

Okay, so we actually have this formula. Letโ€™s use it to find the sum of our first six terms of the geometric series. Well, the first thing we need to do is actually find out what ๐‘Ž and ๐‘Ÿ are. Well, ๐‘Ž is going to be a half because itโ€™s our first term. So weโ€™ve got ๐‘Ž. Now what ๐‘Ÿ going to be? Well, as we said, ๐‘Ÿ is gonna be our common ratio. 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. And that will give us our common ratio.

So Iโ€™m gonna take our second and first terms. So weโ€™re gonna say that ๐‘Ÿ is equal to a quarter which is our second term divided by our first term which is a half. And then, we can say this is equal to a quarter multiplied by two over one because remember if weโ€™re dividing by a fraction, weโ€™re actually find the reciprocal and multiply. So therefore, we can say that ๐‘Ÿ is gonna be equal to two over four which is equal to a half.

Okay, great, weโ€™ve now found our common ratio. Okay, now, we found our ๐‘Ÿ and ๐‘Ž. There is one more variable that we need to know before we substitute into our formula. And thatโ€™s ๐‘›. Well, ๐‘› we said was actually the number of terms. 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.

So now weโ€™ve got all the variables we need to actually substitute back into the formula. So letโ€™s do that and find the sum of our first six terms. So weโ€™re 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. Well, therefore, we can actually divide top and bottom โ€” so the numerator and the denominator by a half.

So therefore, we can say that the sum of the first six terms is gonna be equal to 63 over 64. 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.

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