An Interactive Tutorial by S. Mahaffey (Osborne High School)

Sequences are functions. They are made up of terms that go in a specific order. The position of each term in the sequence is called n. For the 5 th term, n = 5. For the 8 th term, n = 8. Each term is called a n ( pronounced “a sub n” ).

Consider the sequence 2, 4, 6, 8, … The first term is called a 1, “a sub 1”. For this example, a 1 = 2. The second term is 4, so a 2 = 4. The third term is 6, so a 3 = 6. The fourth term is 8, so a 4 = 8. The nth term is called a n.

 Consider the sequence 15, 20, 25, 30, 35, …  What is a 1 ?  What is a 3 ?  What is a 5 ? click

 The domain of a sequence is it’s n values. Since n is just the position of the term, then the domain is the whole numbers 1, 2, 3, 4, etc.  The range of a sequence is it’s a n values, or terms.

Which is the correct domain and range for the sequence 3, 10, 17, 24, … D:{1,2, 3, 4,…} R:{3, 10, 17, 24,…} D:{3, 10, 17, 24,…} R:{1, 2, 3, 4, …} Click here

One more example…  Which is the correct domain and range for the sequence 100, 75, 50, 25, 0, -25, …  D:{100, 75, 50,,25, 0, -25, …}  R:{1, 2, 3, 4, 5, 6, …}  D:{1, 2, 3, 4, 5, 6, …}  R:{100, 75, 50,,25, 0, -25, …} Click here

 Sometimes the terms of the sequence are not listed. Instead, you are given an equation.  If this is the case, you can find the first term by letting n=1, and the second term by letting n=2, and so on.

 Find the domain and range of the sequence a n = 3( n -1) + 2.  First, let’s write out the first 4 terms: a 1 = 3(1-1) + 2 = 3(0) + 2 = 2 a 2 = 3(2-1) + 2 = 3(1) + 2 = 5 a 3 = 3(3-1) + 2 = 3(2) + 2 = 8 a 4 = 3(4-1) + 2 = 3(3) + 2 = 11  So, the domain is {1, 2, 3, 4, …} and the range is {2, 5, 8, 11, …}

 Find the domain and range for a n = 2(n-1) + 10 D: {1, 2, 3, 4, …} R: {2, 4, 6, 8, …} D: {1, 2, 3, 4, …} R: {10, 20, 30, 40, …} D: {1, 2, 3, 4, …} R: {10, 12, 14, 16, …} Click here

 Find the domain and range for a n = -4(n-1) + 1 D: {1, 2, 3, 4, …} R: {-2, -4, -6, -8, …} D: {1, 2, 3, 4, …} R: {1, -3, -7, -11, …} D: {1, 2, 3, 4, …} R: {-4, -8, -16, -32, …} Click here

 Find the domain and range for a n = 5 ∙ 2 (n-1) D: {1, 2, 3, 4, …} R: {5, 10, 20, 40, …} D: {1, 2, 3, 4, …} R: {5, 10, 15, 20, …} D: {1, 2, 3, 4, …} R: {1, 10, 100, 1000, …} Click here

 In the last 3 examples, the sequences were given as formulas and you had to calculate the terms.  Now, we’re going to focus on how to write the formulas when you are given the terms.

 There are two types of equations we’ll be looking at in this tutorial.  The first type of formula is called a recursive formula.  The second type of formula is called a closed formula.

 Recursive formulas are the easiest to write.  Each term is defined by the term that comes before it. (The previous term).  Recursive formulas have 2 parts: define the first term define the next term by its relationship to the previous term.  For example, in the sequence 90, 95, 100, 105, … the first term is 90 and the next term is five plus the previous term.  In the sequence 14, 12, 10, 8, … the first term is 14 and the next term is two minus the previous term.  In the sequence 1, 2, 4, 8, 16, … the first term is 1 and the next term is 2 times the previous term.

English Math Symbol The next termanan The previous terma n-1 Recursive formulas are written almost exactly like you’d say it in English. But instead of writing out phrases like “the next term”, you’ll use these math symbols instead:

 Write the recursive equation for the series 9, 11, 13, 15, …  Identify the pattern: The first term is 9 The next term is equal to the previous term plus 2.  Write the equation: a 1 = 9 a n = a n-1 + 2

 Write the recursive equation for the series 25, 30, 35, 40, 45…  Identify the pattern: The first term is 25. The next term is equal to the previous term plus 5.  Write the equation: a 1 = 25 a n = a n-1 + 5

 Write the recursive equation for the series 10, 9, 8, 7, 6…  Identify the pattern: The first term is 10. The next term is equal to the previous term minus 1.  Write the equation: a 1 = 10 a n = a n-1 - 1

 Write the recursive equation for the series 3, 6, 12, 24, 48…  Identify the pattern: The first term is 3. The next term is equal to the previous term times 2.  Write the equation: a 1 = 3 a n = a n-1 ∙ 2

You try:  Choose the correct recursive formula for the sequence 3, 7, 11, 15, 19, … a 1 = 3a 1 = 3 a n = a n+1 + 4a n = a n a 1 = 4a 1 = 4 a n = a n+1 + 4a n = a n-1 + 4

You try:  Choose the correct recursive formula for the sequence 19, 15, 11, 7, 3, … a 1 = 19a 1 = 4 a n = a n-1 + 4a n = a n a 1 = 19a 1 = -4 a n = a n-1 - 4a n = a n

You try:  Choose the correct recursive formula for the sequence 7, 14, 28, 56, 112, … a 1 = 7a 1 = 7 a n = a n-1 ∙ 2 a n = a n a 1 = 2a 1 = 2 a n = a n-1 + 7a n = a n-1 ∙ 7

You try:  Choose the correct recursive formula for the sequence 1, 8, 64, 512, … a 1 = 8a 1 = 1 a n = a n-1 ∙ 8 a n = a n a 1 = 8a 1 = 1 a n = a n-1 ∙ 1 a n = a n-1 ∙ 8

Recursive formulas are easy to write, but if you want to find the 50 th term, a 50, you have to know the first 49 terms. This can be time consuming. With closed formulas, you can easily find the 50 th term (n = 50). The formula uses the variable n instead of a n-1 (which is the value of the previous term). So, if you want the 10 th term, then n=10. If you want the 20 th term, then n=20…

 When you add or subtract to get to the next term, then you have an arithmetic sequence:  To write a closed formula, start with the recursive formula: a 1 = a n = a n-1 +  Then rearrange the equation to get the closed formulas: a n = (n-1) +

Arithmetic: Recursive a 1 = 3 a n = a n Closed a n = 4 (n-1) + 3

Find the 101 st term of the sequence 2, 6, 10, 14, … First write the recursive formula (The first term is 2 and each additional term is the previous term plus 4) a 1 = 2 a n = a n Next, write the closed formula: a n = 4(n-1) + 2 Finally, find the 101 st term: a 101 = 4(101-1) + 2 = 4(100) + 2 = 402

Find the 51 st term of the sequence 12, 15, 18, 21, … First write the recursive formula (The first term is 12 and each additional term is the previous term plus 3) a 1 = 12 a n = a n Next, write the closed formula: a n = 3(n-1) + 12 Finally, find the 51 st term: a 101 = 3(51-1) + 12 = 3(50) + 12 = 162

Find the 81 st term of the sequence 100, 90, 80, 70, … First write the recursive formula (The first term is 100 and each additional term is the previous term minus 10) a 1 = 100 a n = a n Next, write the closed formula: a n = -10(n-1) Finally, find the 81 st term: a 101 = -10(81-1) = -10(80) = -700

 Write the closed formula for the sequence: a 1 = 3 a n = a n a n = 8(a n-1 ) + 3a n = 3(a n-1 ) + 8 a n = 8(n-1) + 3a n = 3(n-1) + 8

 Write the closed formula for the sequence: a 1 = 15 a n = a n a n = 2(n-1) + 15a n = -2(n-1) + 15 a n = -2(n-1) - 15a n = 2(n-1) - 15

 Write the closed formula for the sequence: 15, 20, 25, 30, … a n = 15(n-1) - 5a n = 15(n-1) + 5 a n = 5(n-1) - 15a n = 5(n-1) + 15

Find the 101 st term of the sequence: 8, 11, 14, 17, 20, … a 101 = 803a 101 = 308 a 51 = 158a 100 = 305

Find the 51 st term of the sequence: 81, 79, 77, 75, 73, … a 101 = -19a 101 = -119 a 51 = -19a 51 = -119

 When you multiply or divide to get to the next term, then you have a geometric sequence:  To write a closed formula, start with the recursive formula: a 1 = a n = a n-1 ∙  Then rearrange the equation to get the closed formulas: a n = ∙ (n-1)

Geometric: Recursive a 1 = 3 a n = a n-1 ∙ 4 Closed a n = 3 ∙ 4 (n-1)

Find the 9 th term of the sequence 1, 2, 4, 8, 16, … First write the recursive formula (The first term is 1 and each additional term is the previous term times 2) a 1 = 1 a n = a n-1 ∙ 2 Next, write the closed formula: a n = 1 ∙ 2 (n-1) Finally, find the 9 th term: a 9 = 1 ∙2 8 = 1∙256 = 256

Find the 10 th term of the sequence 2, 6, 18, 54, … First write the recursive formula (The first term is 2 and each additional term is the previous term times 3) a 1 = 2 a n = a n-1 ∙ 3 Next, write the closed formula: a n = 2 ∙ 3 (n-1) Finally, find the 10 th term: a 10 = 2 ∙ 3 9 = 2 ∙19,683 = 39,366

 Write the closed formula for the sequence: a 1 = 3 a n = a n-1 ∙ 8 a n = 3 ∙ 8 (an-1) a n = 8 ∙ 3 (n-1) a n = 3 ∙ 8 (n-1) a n = 24 (n-1)

 Write the closed formula for the sequence: a 1 = 15 a n = a n-1 ∙ 2 a n = 2 ∙15a n = 15 ∙ 2 (n-1) a n = 30 (n-1) a n = 2(n-1) ∙ 15

 Write the closed formula for the sequence: 5, 15, 45, 135, … a n = 3 ∙5(n-1) a n = 3 ∙5 (n-1) a n = 5 ∙3(n-1) a n = 5 ∙ 3 (n-1)

Find the 8 th term of the sequence: 1, 3, 9, 27, 81, … a 8 = 2,187a 7 = 729 a 8 = 6,561a 7 = 2,187

Find the 10 th term of the sequence: 2, 10, 50, 250, 1250, … a 10 = 1,953,125a 9 = 1,953,125 a 10 = 3,906,250a 9 = 781,250

Oops, wrong answer… Try Again… Remember a 4 will be the fourth term in the sequence, a 2 will be the second term, and a n will be the n th term…

Oops, wrong answer… Try Again… Remember, the domain is the set of whole numbers… The range is the terms of the sequence…

Oops, wrong answer… Try Again… Remember, a 1 is the first term. plus something a n = a n-1 minus something times something divided by something

Oops, wrong answer… For arithmetic sequences, where you add or subtract to get to next term: Recursive formula:a 1 = a n = a n-1 + Closed formula: a n = (n-1) +

Oops, wrong answer… For geometric sequences, where you multiply or divide to get to next term: Recursive formula:a 1 = a n = a n-1 ∙ Closed formula: a n = ∙ (n-1)