Recursive Formula for Arithmetic Sequences

Arithmetic Sequences and Recursive Formulas

by Tess Schuck

Multiple Choice

Find the next three terms:
5, 8, 11, 14...

• 16, 19, 22
• 17, 19, 21
• 17, 20, 23
• 16, 18, 20 

Multiple Choice

The next two terms of the sequence 2,-4, 8,-16 are

• 32, 64

• 32, -64

• -64, 108

• 32, 20

Poll

How confident are you that you can find the common difference of a sequence?

• I can do it every time!

• I can do it if there are no fractions/decimals

• I can do it sometimes if it is just positive

• I don't know how to find the common difference

​Common Difference

​The common difference is a fancy way of saying how much you grew - it is like your slope!

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​ex: 1, 2, 3, 4, 5... what would be the next number? How do you know?

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​Finding common difference

​If you can't find the common difference easily by adding, try using subtraction!

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​ex/ 1.4, 1.8, 2.2, 2.6, 3, 3.4 ...

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​Take the first two numbers and SUBTRACT to find the difference

​1.8-1.4 = +0.4 = difference

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​Check if every time you go up by the same amount (0.4)

​Recursive Formula

​A recursive formula uses previous terms to find the next term. We need two things:

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​1. Your starting number (we call this a₁)

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​2. Your common difference (we call this d)

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​Ex: 11, 13, 15, 17, 19, 21... a₁ = d=

​Recursive Formulas

​To write a recursive formula, you just have to fill in the a₁ and d in the equation with the proper numbers!

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​a_n = a_n-1 + d

​a₁ = ?

​Example 1

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• ​a₁ = 12

• ​d = 4 (or positive 4)

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​The sequence starts with 12 and goes up 4 each time

​ 12, 16, 20, 24, 28, 32...

​RECURSIVE FORMULA

​

​a₁ = 12

​a_n = a_n-1 + 4

Multiple Choice

Choose the sequence that matches this recursive formula:

a₁ = 13

a_n=a_n-1 + 2

• 13, 15, 17, 19, 21...

• 2, 13...

• 2, 15, 28, 41., 54..

• 13, 26, 39, 52...

​​RECURSIVE FORMULA

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​a₁ = 13

​a_n = a_n-1 +2

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​This problem says a₁ the START is 13. This had to be the first number.

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​It says d (the common difference) is 2. This has to be how much you add.

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​13, 15, 17, 19... Is your answer!

Multiple Choice

Which is the recursive formula for this sequence?

20, 15, 10, 5, 0, -5...

• a₁ = 20

  a_n=a_n-1 + 5

• a₁ = 20

  a_n=a_n-1 - 5

• a₁ = -5

  a_n=a_n-1 + 20

• a₁ = -5

  a_n=20 -5

​What is the THIRD number in the sequence if your recursive formula is

a₁ = 9

a_n = a_n-1 - 4

​Try writing out the sequence!

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​First number is 9

​Then the common difference is MINUS 4.

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​9-4 = 5. That is the second number. So far the sequence is 9, 5...

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​Then the next number will be 5 - 4 = 1

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​9, 5, 1.... The third number in the sequence is 1!

Multiple Choice

Identify the common difference of the given arithmetic sequence.

5, 1, -3, -7, ...

• d = -4

• d = -5

• d = 5

• d = 7

Multiple Select

Which of the the following are examples of arithmetic sequence?

• $$\frac{1}{4},\frac{1}{2},\frac{3}{4},1$$

• $$6,11,16,21,...$$

• $$100,95,90,85,80,75$$

• $$100,50,25,\ ...$$

Multiple Choice

Find the common difference.

• $$d=4$$

• $$d=6$$

• $$d=8$$

• $$d=5$$

Multiple Choice

Find the common difference.

• $$d=-3$$

• $$d=3$$

• $$d=-2$$

• $$d=-4$$

Multiple Choice

Write the recursive formula for 6, 17, 28, 39, 50....

• a_n=a_n-1+11
  a₁=6
• a_n=a_n-1-4
  a₁=6
• a_n=a_n-1+9
  a₁=6
• a_n=a_n-1+6
  a₁=6

Multiple Choice

Given the recursive formula, find the first four terms:
a_n = a_n-1 + 5
a₁ = -16

• -16, -21, -26, -31
• 5, -11, -27, -43
• -16, -80, -500, -200
• -16, -11, -6, -1

Multiple Choice

Find the recursive formula. $$-4,-12,-20,-28,\ ...$$  

Multiple Choice

Given the recursive formula below, find the first five terms.

u₁ = 3

u_n = u_n-1 + 2

• 3, 5, 7, 9, 11

• 5, 7, 9, 11, 13

• 2, 4, 6, 8, 10

• 1, 3, 5, 7, 9

Multiple Choice

Find the first five terms.

• $$-8,-17,-26,-35,-44$$  

• $$-18,-26,-34,-42,-50$$  

• $$-18,-28,-38,-48,-58$$  

• $$-8,-18,-28,-38,-48$$