The term-to-term rule of sequence is add five and then multiply by two. Part a) says, if the first term of the sequence is two, calculate the third term of the sequence. Part b) says, Jess takes the first four terms of the sequence and reverses their order to create a new sequence. What is the term-to-term rule of Jess’s new sequence?
Starting with part a, we have the first term as two. And then, we’ll follow the rule, add five and then multiply by two. To two, we add five, and whatever that value is, we multiply by two. And this will give us our second term. Two plus five is seven, and seven times two is 14. Because this is a term-to-term sequence, we’ll take our second term 14 and we’ll use it to begin the process again. So, we need to add five to 14 and then multiply that by two. This is what produces our third term. 14 plus five is 19, and 19 times two equals 38. The third term in this sequence is 38.
If we now, for part b, want to create a rule that reverses this order, we need a rule where we start with 38 and we end up with 14. We can start by dividing 38 by two. This is 19. And if we subtract five from 19, we get 14. Notice how the last thing we did in the first sequence was multiply by two. To reverse that, the first thing we need to do is divide by two. The first thing we did in the beginning of the sequence was add five. And that now is reversed to say subtract five.
We don’t need brackets here because division comes before subtraction in the order of operations. We’re pretty sure that we need to divide by two and then subtract five. But to check, we can apply our same rule to see if we do this to 14 and it produces two. Is 14 divided by two minus five equal to two? 14 divided by two is seven, and seven minus five equals two. This proves that our new rule reverses the order of the first term-to-term sequence. And we write out this rule to say divide by two and subtract five.