The first five terms of an arithmetic sequence are shown in the table below. Five, 14, 23, 32, and 41. Part a) Write an expression for the 𝑛th term of the sequence in terms of 𝑛. Part b) The 𝑛th term of a different arithmetic sequence is three 𝑛 plus 𝑐. The 21st term in this sequence is 19. What is the value of the constant 𝑐?
So the first thing to look at in this question is the fact that it’s an arithmetic sequence. What does this mean? Well, an arithmetic sequence is a sequence that has a common difference between each term. So therefore, the difference between term one and term two will be the same as difference between term two and term three and so on.
And we can actually check that with our sequence. And when we do, we can see that yes this is correct because actually there’s a common difference of nine. Because to get from five to 14, you add nine because five to 10 is five add another four is nine. So that gets us to 14. Then, again from 14 to 23, well, you need to add six to get to 20 plus another three so that’s add nine again. 23 to 32, you add seven to get to 30, then add another two to get to 32 so again add nine. Then, finally, from 32 to 41, you add eight to get to 40 plus another one to get to 41 is add nine.
Okay, but how are we gonna find the 𝑛th term of this sequence? So to start off with, we know that the beginning of our 𝑛th term is going to be nine 𝑛. And that’s because we have nine, well it’s positive nine because we added nine each time, and that’s our common difference and then 𝑛 which means the term number. And you always put the common difference before 𝑛 when we’re actually looking to form our 𝑛th term.
But is this our finished 𝑛th term? Well, no, it’s not. Because if we actually worked out nine 𝑛, where 𝑛 is the term number for our first five terms, we’d get nine multiplied by one which gives us nine, nine multiplied by two which gives us 18, nine multiplied by three which gives us 27, nine multiplied by four which is 36, and nine multiplied by five which would be 45. And these values don’t match our sequence.
So what I’ve done is actually written our sequence below. And we need to see well what do we need to do to actually get from the values given by nine 𝑛 to the value we want for that term number in our sequence. So to get from nine to five, we’d have to subtract four and 18 to 14, again subtract four, 27 to 23, subtract four, and so on.
So, therefore, the 𝑛th term is going to be nine 𝑛 minus four as nine 𝑛 minus four will actually give us each of the terms in our sequence.
And we can do a quick check just to make sure this works okay. So I’m gonna choose the fourth term. So if I’m gonna check with the fourth term, then we’re gonna substitute four instead of 𝑛. So we’re gonna get nine multiplied by four minus four which is gonna give us 36 minus four which is equal to 32, which is in fact our fourth term of our sequence.
Okay, great, now, let’s move on to part b. So in part b, we’re told that actually the 𝑛th term of a different arithmetic sequence is three 𝑛 plus 𝑐. And we’re also told that the 21st term is 19. So in order to actually find out what 𝑐 is, so the constant 𝑐, let’s write down some other information that we can glean from the information in the question.
Well, first of all, we know that 𝑛 is going to be 21 and that’s because we’re interested in the 21st term. So therefore, if we actually substitute 𝑛 is equal to 21 into three 𝑛 plus 𝑐, we’re told that the value is going to be 19 and that’s because the 21st term in the sequence is 19.
So therefore, we can say that three multiplied by 21 because that’s the 𝑛 that we substituted in plus 𝑐 is going to be equal to 19. So therefore, we’re gonna get 63 plus 𝑐 is equal to 19. So therefore, to find 𝑐, what we need to do is subtract 63 from both sides of the equation. And that’s because 63 minus 63 is zero. So it leaves 𝑐 on its own on the left-hand side. But whatever you do to one side of the equation, you must do to the other side.
So then we get 𝑐 is equal to negative 44.