Explicit and Recursive Rules

Question

a. Find the 1,000th term of the sequence 3, 8, 13, 18, 23,…

b. Which rule did you use to find the 1,000th term? Why did you choose that rule?

c. What is a disadvantage of a recursive rule when compared to an explicit rule?

Answer 1

This sequence is an arithmetic sequence because we add 5 to go from one term to the next.

For me, I find it easiest to use the linear equation y = mx + b to form the equation for an arithmetic sequence since it is something you are already familiar with. Using y = mx + b, "y" is the term value of the "x" term, meaning since "3" is the 1st term, "3" is "y" when "1" is "x" or since "8" is the 2nd term, "8" is the "y" when "2" is the "x". The "m" is just the number you are adding to each term, so in this case m = 5. The only thing that requires a LITTLE effort is the "b". The "b" (usually the y intercept) is the value when x = 0. You are not given a "zeroth" term, only a 1st term. But you can easily go backwards 5 from the first term to get the "zeroth" term; in this case that would be -2 because 3 - 5 = -2.

So the equation for the sequence rule is y = 5x - 2. Try it for the 3rd term (x = 3):

y = 5(3) - 2

y = 15 - 2

y = 13

So it works.

Usually, for sequences they use different variables than x and y though, The term value is usually called "a_n" (instead of "y") and the term number is usually called "n" instead of "x". So the equation could be:

a_n = 5n - 2

This is called an explicit rule.

a_n confuses people sometimes, but it is a variable that changes as you go. For the first term (n = 1), it's called a₁ (because the n is 1). For the 2nd term (n = 2), it is a₂, etc.

To find the 1000th term (or a₁₀₀₀), plug in n = 1000 into the explicit rule:

a_n = 5n - 2

a₁₀₀₀ = 5(1000) - 2

a₁₀₀₀ = 5000 - 2

a₁₀₀₀ = 4998

Another way of generating the sequence is to use the recursive rule. The recursive rule works like your brain does when someone asks you what the next number in the sequence 3, 8, 13, 18, 23,… is. You just add 5 to the last number and tell them "28". So the recursive rule is:

The "next term" is "the current term" plus 5

If we call the current term "a_n", then the "next term" would be "a_n+1" which would make the equation:

The "next term" is "the current term" plus 5

a_n+1 = a_n + 5

HOWEVER, you must also give direction about where to start. So you must say a₁ = 3

Now, given a_n+1 = a_n + 5 and a₁ = 3, you can generate the sequence.

a₁ = 3

a₂ = a₁ + 5 = 3 + 5 = 8

a₃ = a₂ + 5 = 8 + 5 = 13 etc.

Although this could be done to find the 1000th term, it would take all day.