Video Transcript
Find the first five terms of the sequence that follows the rule nine ๐ plus four, where ๐ represents the position of a term in the sequence.
So weโve been given the general rule or ๐th term for calculating any term in this sequence. Itโs nine ๐ plus four. And we know that ๐ represents the position of a term in the sequence, which is also sometimes called the order of a term. For example, for the first term in the sequence, ๐ is equal to one. For the second term, ๐ is equal to two, and so on. We can find the first five terms of this sequence by substituting the relevant values of ๐ into the ๐th term formula. For example, for the first term, ๐ is equal to one. So substituting ๐ equals one into the rule nine ๐ plus four, we have nine multiplied by one plus four. Thatโs nine plus four, which is equal to 13. And so this is the first term in the sequence.
We can then calculate the second term by substituting ๐ equals two into the general rule. Nine multiplied by two plus four, thatโs 18 plus four, which is equal to 22. We can continue in this way. For the third term, we substitute ๐ equals three. Nine multiplied by three plus four, thatโs 27 plus four, which is equal to 31. For the fourth term, ๐ is equal to four. So substituting into the general rule, we have nine multiplied by four, thatโs 36, plus four, which is equal to 40. And finally, the fifth term, nine multiplied by five plus four, thatโs 45 plus four, which is equal to 49. So we found that the first five terms of this sequence are 13, 22, 31, 40, and 49.
There is another way to answer this question, which is to recognize that the general rule weโve been given, nine ๐ plus four, is in the form ๐๐ plus ๐ and therefore this sequence is whatโs known as an arithmetic sequence. Terms in an arithmetic sequence have a common difference, which means the terms always increase or decrease by the same amount and that increase or decrease is the same as the coefficient of ๐ in the general rule. In our rule of nine ๐ plus four, the coefficient of ๐ is nine. And so the common difference of the sequence is nine, which means that the terms increase by nine each time.
Once weโve calculated the first term in the sequence then, which we did by substituting ๐ equals one to give 13, we can calculate subsequent terms by adding nine. 13 plus nine gives 22, which we can see is the same as what we calculated the second term of this sequence to be. Adding nine again, 22 plus nine gives 31, which we can see is the same as what we calculated the third term of the sequence to be. We could continue in this way to see that, once again, the fourth and fifth terms are 40 and 49. Whichever method we choose, we of course arrive at the same answer. The first five terms of the sequence that follows the rule nine ๐ plus four are 13, 22, 31, 40, and 49.
