The first four terms of a sequence are shown in the table, negative one, negative four, negative seven, negative 10. Circle the expression for the 𝑛th term of the sequence. Is it three 𝑛 minus 10? Is it three 𝑛 minus one, two minus three 𝑛, or 𝑛 minus three.
This is an arithmetic or a linear sequence. And we know this because it has a common first difference. Let’s find the value of that first difference. What do we do to get from negative one to negative four? We subtract three. What do we do to get from negative four to negative seven. Once again, we subtract three. And to get from negative seven to negative 10, we subtract three. This is the common first difference. It’s negative three.
The common difference tells us the coefficient of 𝑛. That’s the number of 𝑛s we have. So here, the number of 𝑛s we have is negative three. And the first part of our 𝑛th term is negative three 𝑛. But how do we find out the other part? Well, one way of doing this is to list the negative three 𝑛 sequence above the sequence we’ve been given. The first term of the sequence negative three 𝑛 is when 𝑛 is one. That’s negative three multiplied by one, which is negative three.
The second term is when 𝑛 is two. That’s negative three multiplied by two, which is negative six. The third term is found by multiplying negative three by three, which is negative nine. And that means that the final term in the sequence given by negative three 𝑛 is negative 12.
We need to work out what we can add to the sequence negative three 𝑛 to get to the sequence in our question. Well, to get from negative three to negative one, we add two. To get from negative six to negative four, we add two and so on. So our 𝑛th term is negative three 𝑛 plus two. This is the same as saying two minus three 𝑛. So the expression for the 𝑛th term of the sequence is two minus three 𝑛.
Now, a nice little method we can use to find the 𝑛th term for any arithmetic sequence is to remember these three letters, Dno. Now, this obviously doesn’t mean that we don’t know how to find the 𝑛th term. Each letter stands for something. The D stands for difference. We find the common difference. The 𝑛 stands for 𝑛. Whatever the common difference is, we multiply it by 𝑛. That’s what we did for the first part before. The o stands for the zero term. And this is where this method slightly differs.
The zero term is the number that would become before the first term in our sequence. Our sequence is going down by three each time. So to find the zero term, we need to add three. Negative one add three is two.
So once again, we have shown that the 𝑛th term of this sequence is negative three 𝑛 plus two or two minus three 𝑛.