# Question Video: Using the General Term of a Sequence to Calculate a Given Term Mathematics

If the fourth term of the sequence (ππ + π) is 26 and the eighth term is 46, find the ninth term of this sequence.

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### Video Transcript

If the fourth term of the sequence ππ plus π is 26 and the eighth term is 46, find the ninth term of this sequence.

Weβve been given a formula for calculating the general term in this sequence. We could write this as π sub π equals ππ plus π, where π sub π represents the πth term. We want to calculate the ninth term in this sequence, which means we need to substitute the value π equals nine into this general formula. But the trouble is, we donβt yet know what the values of π and π are. Weβll need to work these out first, which we can do using the other information in the question.

Weβre told that the fourth term in this sequence is 26 and the eighth term is 46. This means that when we substitute π equals four into this general formula, we get 26, and when we substitute π 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 π plus π equals 26. And if the eighth term is 46, we have the equation eight π plus π equals 46.

We can now solve this system of linear simultaneous equations to determine the values of π and π. If we subtract the first equation from the second, then the π-terms cancel one another out, leaving four π is equal to 20. We can divide both sides of this equation by four, and we find that the value of π is five. We can then substitute this value of π back into either equation, Iβve chosen equation one, to find the value of π. We have four multiplied by five plus π is equal to 26. That simplifies to 20 plus π equals 26. And subtracting 20 from each side, we find that the value of π is six.

So having calculated the values of π and π, we can now write down this general rule explicitly. And we have that the πth term of this sequence is equal to five π plus six. To calculate the ninth term, we substitute π equals nine, giving five multiplied by nine plus six. Thatβs 45 plus six, which is equal to 51. So by determining the values of π and π, 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 πth term rule π π equals ππ plus π, the first term π one is π plus π. The second term π two is two π plus π. The third term is three π plus π. The fourth term is four π plus π. So we observe that to get from one term to the next, we add the value of π. Once we determined the value of π is five, we could calculate the ninth term by adding five to the eighth term. So weβd have π nine is equal to π eight plus five; thatβ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 π sub π plus one is equal to five plus π sub π. Using either method, we find that the ninth term of the sequence with general term ππ plus π, where the fourth term is 26 the eighth term is 46, is 51.