Evaluate the sum of the terms from 𝑟 equals three to 𝑟 equals 22 for three 𝑟 plus four.
Can’t wait to start to look at this kind of question. This kind of notation can sometimes catch students out; they’re not quite sure what it means. But the first sign here 𝛴; this means the sum of. And this value at the bottom of the 𝛴 is telling us what it wants the first value of our variable to be, so in this case our first value of 𝑟 is equal to three. And the 22 at the top of the 𝛴, this tells us what it wants the final value of our variable to be. Great! So now we know what it means, we can actually solve the problem.
To solve this problem, we need to use this formula. So we’ve got the sum of 𝑛 terms is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one 𝑑. Okay, so let’s have a look at what these actual letters mean. We got 𝑎, which is our first term, 𝑑, which is our common difference, so the difference between each term, and 𝑛, which is our number of terms.
Okay, so now we know the formula and what each of the variables are, we’re gonna start to solve the problem. In order to do that, we first of all need to calculate 𝑎, 𝑑, and 𝑛. So we’re gonna start with 𝑎. Well, 𝑎 is our first term, so we can say that 𝑎 is when 𝑟 is equal to one. So now we can calculate what 𝑎 will be cause we substituted 𝑎 into three 𝑟 plus four, which gives us that 𝑎 is equal to seven.
Okay, great! So we’ve now found 𝑎. Now, we can move on to find 𝑑. To find 𝑑, we already know that 𝑎 one, our first term, is seven. What we need to do now is find 𝑎 two, our second term, so we can work out the common difference between the two of them. And by doing that, we actually substitute 𝑟 is equal to two into three 𝑟 plus four, which gives us that 𝑎 two, our second term, is equal to 10. And now, we can just find the difference between the two terms, which is 10 minus seven, which is equal to three.
Okay, great! So we’ve now found the common difference. We found that by finding the second term and the first term and then subtracting the first term from the second term, but actually, we could have done it without having to work that out at all because if we look at our expression three 𝑟 plus four, we can see that our coefficient of 𝑟 is actually three. And in this kind of situation, the coefficient of 𝑟 or whichever the letter the variable is, that will tell you what the common difference is because that’s what you’re gonna multiply that term number by each time.
So now, I’m gonna work out sum of the first 22 terms, so if you look at it, it says 22 terms. We’re gonna have a look at that. So we now know that the 𝑛 is going to be equal to 22 cause we’re looking at the sum of the first 22 terms. So we can substitute our other values in. So we have sum of the first 22 terms equal to 22 divided by two and then we’ve got two times seven because 𝑎 is equal to seven plus 22 minus one because 𝑛 is 22 multiplied by three cause 𝑑 is equal to three. Great! Okay, so we can now work this out, which gives us 11 multiplied by 77. So therefore, the sum of the first 22 terms is equal to 847.
Great! So now we’ve solved the problem, haven’t we? Well, actually, no, and this is where the most common mistake is. We found the sum of the first 22 terms, but if we look back at our original question, it says what is the sum of the terms from 𝑟 is equal to three until 𝑟 is equal to 22. So we want the terms from three to 22. So what we need to do is actually do one more step because the sum of the terms from three to 22 is equal to the sum of the first 22 terms minus the sum of the first two terms because that would leave us with the third to the 22nd term.
Therefore, we can say that the sum of the terms from 𝑟 is equal to three to 𝑟 is equal to 22 of three 𝑟 plus four is equal to 847 minus 10 plus seven, and it’s 10 plus seven because it says it’s the sum of the first two terms. We previously worked out the first two terms because we got the first term is equal to seven and the second term is equal to 10. So we add those together to get the sum, which gives us 847 minus 17.
You could have also found the sum of the first two terms using the formula above and substituting 𝑛 is equal to two. And now, we’ve reached our final answer because we’ve got 847 minus 17 gives us 830. So we can say the sum of the terms from three to 22 of three 𝑟 plus four is equal to 830.