The grid shows the first five terms in an arithmetic sequence. Negative five, two, nine, 16, and 23. Find an expression for the 𝑛th term of the sequence, in terms of 𝑛.
Remember an arithmetic sequence is one for which the consecutive terms have a common difference. To find the 𝑛th term which is an algebraic rule to help us find any term in the sequence, we first need to find this common difference.
Let’s see what we need to do to get from one term to the next. To get from negative five to two, we must add seven. To get from two to nine, we also add seven. To get from nine to 16, we add seven. And you’ve guessed it; to get from 16 to 23, we also add seven.
The common difference in this sequence then is seven. The common difference always tells us the coefficient of 𝑛; that’s how many 𝑛s we have. So the 𝑛th term for this sequence begins seven 𝑛. To find the other part, we list out the first five terms of this sequence seven 𝑛; that’s basically the seven times tables.
The first number in the sequence given by seven 𝑛 is found when 𝑛 is one. So it’s seven multiplied by one which is of course seven. The second term in the sequence is when 𝑛 is two. So it’s seven multiplied by two which is 14. We then have seven multiplied by three which is 21, seven multiplied by four which is 28, and seven multiplied by five which is 35.
To find the other part of our 𝑛th term rule, we need to work out how we get from the sequence seven 𝑛 to the sequence in our question, in other words, how do we get from seven to negative five and 14 to two, 21 to nine, and so on.
In fact to get from each term in the seven times tables to our sequence, we subtract 12. So we can say that the expression for the 𝑛th term of this sequence is seven 𝑛 minus 12.
There is an easier way to remember this. If someone asks you how to find the 𝑛th term of an arithmetic sequence, you can say DNO. Now, of course, this doesn’t actually mean you don’t know. Each part of the mnemonic stands for something.
The D stands for difference. We find the difference between each term. The N simply stands for 𝑛; we multiply that difference by 𝑛. The O stands for add the zero term; this is the term that will come before the first term in the sequence. Let’s look at how this works for our sequence.
We already showed that the common difference is seven. So the D part of our mnemonic is seven. And we said that we take the difference and multiply it by 𝑛. So actually, the first part of our sequence is seven 𝑛. The zero term is the term before the first time.
Since our first term is negative five and to get to the next term in the sequence we add seven, we can find the zero term by going backwards and subtracting seven. If we start at negative five and then take seven away, we go down the number line and we end up at negative 12. When we add this zero term, we get seven 𝑛 plus negative 12, which we know we can simply write as seven 𝑛 minus 12.
We have shown that the expression for the 𝑛th term of this sequence in terms of 𝑛 is seven 𝑛 minus 12.