If the fourth term of the sequence 𝑎𝑛 plus 𝑏 is 26 and the eighth term is 46, find the ninth term of this sequence.
We’ve been given a formula for calculating the general term in this sequence. We could write this as 𝑇 sub 𝑛 equals 𝑎𝑛 plus 𝑏, where 𝑇 sub 𝑛 represents the 𝑛th term. We want to calculate the ninth term in this sequence, which means we need to substitute the value 𝑛 equals nine into this general formula. But the trouble is, we don’t yet know what the values of 𝑎 and 𝑏 are. We’ll need to work these out first, which we can do using the other information in the question.
We’re told that the fourth term in this sequence is 26 and the eighth term is 46. This means that when we substitute 𝑛 equals four into this general formula, we get 26, and when we substitute 𝑛 equals eight, we get 46. We can therefore write down two simultaneous equations. If the fourth term is 26, then we have the equation four 𝑎 plus 𝑏 equals 26. And if the eighth term is 46, we have the equation eight 𝑎 plus 𝑏 equals 46.
We can now solve this system of linear simultaneous equations to determine the values of 𝑎 and 𝑏. If we subtract the first equation from the second, then the 𝑏-terms cancel one another out, leaving four 𝑎 is equal to 20. We can divide both sides of this equation by four, and we find that the value of 𝑎 is five. We can then substitute this value of 𝑎 back into either equation, I’ve chosen equation one, to find the value of 𝑏. We have four multiplied by five plus 𝑏 is equal to 26. That simplifies to 20 plus 𝑏 equals 26. And subtracting 20 from each side, we find that the value of 𝑏 is six.
So having calculated the values of 𝑎 and 𝑏, we can now write down this general rule explicitly. And we have that the 𝑛th term of this sequence is equal to five 𝑛 plus six. To calculate the ninth term, we substitute 𝑛 equals nine, giving five multiplied by nine plus six. That’s 45 plus six, which is equal to 51. So by determining the values of 𝑎 and 𝑏, we found that the ninth term of this sequence is 51.
An alternative approach we could have taken is to observe that there is in fact a pattern in the terms of this sequence. Using the 𝑛th term rule 𝑇 𝑛 equals 𝑎𝑛 plus 𝑏, the first term 𝑇 one is 𝑎 plus 𝑏. The second term 𝑇 two is two 𝑎 plus 𝑏. The third term is three 𝑎 plus 𝑏. The fourth term is four 𝑎 plus 𝑏. So we observe that to get from one term to the next, we add the value of 𝑎. Once we determined the value of 𝑎 is five, we could calculate the ninth term by adding five to the eighth term. So we’d have 𝑇 nine is equal to 𝑇 eight plus five; that’s 46 plus five, which is again equal to 51.
In fact, we could use this relationship between successive terms to write down a recursive definition of the sequence. To get from one term to the next, we add five. So we can say that 𝑇 sub 𝑛 plus one is equal to five plus 𝑇 sub 𝑛. Using either method, we find that the ninth term of the sequence with general term 𝑎𝑛 plus 𝑏, where the fourth term is 26 the eighth term is 46, is 51.