The sum of the terms of a sequence is called a series. A geometric series is the sum of a geometric sequence. A geometric series with 𝑛 terms can be written as 𝑆 𝑛 is equal to 𝑎 plus 𝑎𝑟 plus 𝑎𝑟 squared plus 𝑎𝑟 cubed, and so on until you reach 𝑎𝑟 𝑛 minus one, where 𝑎 is the first term and 𝑟 is the common ratio, and the common ratio is the number you multiply one term by to get to the next term in the sequence, but 𝑟 cannot be one.
Before we look at the question, we’re actually gonna have a look at this statement that 𝑟 cannot be equal to one. The common ratio cannot be equal to one because if it was, you’d have a sequence that looks like this, which means that actually if you had 𝑎, 𝑎 times one, 𝑎 times one squared, 𝑎 times one cubed, it would always be 𝑎, so every term would be 𝑎, so actually it would not be a sequence; it’d just be a repetition of a number.
Okay, now let’s look at the last part of the question: find the sum of the first six terms of a geometric series with 𝑎 is equal to 24 and 𝑟 is equal to a half. To enable us to solve this problem, we need to use this formula, which tells us that the sum of the first 𝑛 terms is equal to 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 over one minus 𝑟, again with the parameter that 𝑟 cannot equal to one.
And we discussed one reason why it wouldn’t work if you’re looking at each term of a sequence, but also if you look at the equation. The denominator one minus 𝑟 would be equal to zero, and obviously we can’t have that because it wouldn’t give us a real solution.
When solving a problem like this, we’re gonna use a formula. I like to write down the values we have and those we’re looking for. So first of all, we know that 𝑎 is equal to 24, so our first term is 24. And we also have 𝑟 is equal to a half, so we know that the common ratio is equal to a half. We’ll also need to find 𝑛, as you can see is in our formula, so 𝑛 is equal to six because this is the number of terms.
And if we look at our question, it says the first six terms. And finally, the sum of the first six terms, which we write as 𝑆 with six, is what we’re looking to find in the question.
Now we have our values; we can substitute them into the formula to find the sum of the first six terms, which gives us 𝑆 six or the sum of the first six terms is equal to 24 multiplied by one minus a half to the power of six over one minus a half.
We’re gonna simplify this, which gives us 189 over eight divided by a half, which gives us 189 over four, cause remembering, quick tip, dividing by a half is the same as multiplying by two. Then finally simplify it by converting into a mixed number; it gives us the sum of the first six terms of the geometric series is 47 and a quarter.
Again the reason that’s 47 and a quarter is that four goes into 189 47 times with one leftover, so it gives us 47 and a quarter.