Video Transcript
Find the common difference of the arithmetic sequence given the general term 𝑎 sub 𝑛 is equal to negative three 𝑛 minus one.
Remember, an arithmetic sequence is one in which the difference between consecutive terms is constant. And it’s this common difference that we’re asked to find in this question. There are two ways that we could do this. The first method is we could use the formula for the general term 𝑎 sub 𝑛 equals negative three 𝑛 minus one to calculate some of the terms in this sequence and then look at what the common difference is.
For example, for the first term in the sequence, 𝑛 is equal to one. So substituting, we have 𝑎 sub one is equal to negative three multiplied by one minus one. That’s negative three minus one, which is negative four. For the second term, 𝑛 is equal to two. So we have 𝑎 sub two is equal to negative three multiplied by two minus one. That’s negative six minus one, which is equal to negative seven. And to be on the safe side, let’s calculate one more. For the third term, 𝑛 is equal to three. So we have negative three multiplied by three minus one. That’s negative nine minus one, which is negative 10.
We’ve, therefore, calculated the first three terms in this sequence. And indeed, we could’ve calculated any successive terms by substituting any values of 𝑛. Let’s now look at these three values to determine the common difference. To get from negative four to negative seven, we have to subtract three. And to get from negative seven to negative 10, we also have to subtract three. This tells us that to get from one term to the next throughout our arithmetic sequence, we subtract three. And therefore, the common difference is negative three.
The other way we could answer this question without calculating any terms is to use the general term we’ve been given. And this requires a little bit more familiarity with arithmetic sequences. The general term of an arithmetic sequence can always be expressed in the form 𝑏𝑛 plus 𝑐. That’s some multiple of 𝑛 plus a constant. It’s always the case that the coefficient of 𝑛 in the general term gives the common difference of the sequence. The general term for our sequence is negative three 𝑛 minus one. And so we see that the coefficient of 𝑛 is negative three.
Using two different methods then, we’ve found that the common difference of the arithmetic sequence with a general term negative three 𝑛 minus one is negative three.
