Which of the following expressions can be used to find the 𝑛th term of the given sequence, where 𝑛 represents the position of a term in the sequence? The options are 𝑛 plus two, three 𝑛 squared plus two, 𝑛 squared plus two, three 𝑛 plus two, or three 𝑛.
Let’s begin by looking at our sequence more closely, the terms of which are the second row in the table we’ve been given. To get from the first term to the second, that’s from five to eight, we need to add three. To get from the second term to the third, that’s from eight to 11, we also add three. The same is true for the remaining two terms in the sequence. This means that our sequence has a constant common difference or a constant common first difference. And it is therefore a linear sequence.
A linear sequence has a general term of 𝑎𝑛 plus 𝑏, where 𝑎 and 𝑏 are constants. We can therefore rule out two of the options we’ve been given, three 𝑛 squared plus two and 𝑛 squared plus two, because these involve an 𝑛 squared term. These are the 𝑛th term rules for quadratic sequences rather than linear sequences. The value of 𝑎 in this 𝑛th term rule or general term is the common difference between the terms in the sequence. So the value of our common difference is three which means that our 𝑛th term rule has the form three 𝑛 plus 𝑏.
Another way of thinking about this is that if our sequence goes up by threes each time, then it’s related to the three times table, which has the general term three 𝑛. We can therefore rule out one more option that we were given, 𝑛 plus two, because it has the wrong coefficient of 𝑛, although it does describe a linear sequence, just not this one.
There are two ways to work out the value of 𝑏 in our 𝑛th term rule. The first method is to write down the terms in the sequence, three 𝑛, and then see what we need to add or subtract to this sequence in order to give our sequence. The first term in the sequence three 𝑛 is three multiplied by one, which is three. The second term is three multiplied by two, which is six, then three multiplied by three, nine, three multiplied by four, which is 12, and then three multiplied by five, which is 15. Comparing this sequence, three 𝑛, with our sequence, we see that we have to add two each time to three 𝑛 in order to give the terms of our sequence. So the value of 𝑏 is positive two.
Another way to find the value of 𝑏 is to think of it as the zero term in the sequence. That means the sequence with term number zero which comes before the first term in the sequence. If we have to add three to find the next term in the sequence, then to find the previous term, going back the other way, we need to subtract three. So to work out the zero term, we need to subtract three from the first term of five. Five minus three is two. So, again, we see that the value of 𝑏 is two. The 𝑛th term of the given sequence then is three 𝑛 plus two.
We can check this by substituting any value of 𝑛 and confirming that we do get the correct value for that term. For example, when 𝑛 is equal to four, substituting into our 𝑛th term rule gives three multiplied by four plus two. Three multiplied by four is 12. And adding two gives 14. We see in the table that this is the correct value for the fourth term. And therefore this confirms that our 𝑛th term rule of three 𝑛 plus two is correct.