Recursive Formula for Arithmetic Sequence

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

​

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

Fill in the Blanks

What is the common difference in this sequence?

7, 10, 13, 16, 19...

Fill in the Blanks

What is the common difference in this sequence?

10.5, 8.9, 7.3, 5.7 ...

​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=

Fill in the Blanks

11, 13, 15, 17, 19, 21...

Find a₁ =

Fill in the Blanks

11, 13, 15, 17, 19, 21...

Find 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!

​

​a_n = a_n-1 + d

​a₁ = ?

​Example 1

​

​

• ​a₁ = 12

• ​d = 4 (or positive 4)

​

​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

​

​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

Fill in the Blanks

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

a₁ = 9

a_n = a_n-1 - 4

​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!

​

​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?

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

• $$6,11,16,21,...$$

• $$100,95,90,85,80,75$$

• $$100,50,25,\ ...$$

Fill in the Blanks

What is the common difference of the arithmetic sequence 18,24,30,36, ...?

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.

• $$-8,-17,-26,-35,-44$$  

• $$-18,-26,-34,-42,-50$$  

• $$-18,-28,-38,-48,-58$$  

• $$-8,-18,-28,-38,-48$$