Recursive Sequences

Recursive and Explicit Sequences

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Recursive Formula

Allows you to find any term using the PREVIOUS term.

must state what the first term (a₁) equals

ends with n>2

Recursive Formulas

• arithmetic (linear)- the previous term PLUS the common difference

• a_n=a_n-1+d

• geometric (exponential) - the previous term TIMES the common ratio

• a_n=r * a_n-1

Recursive Example Find the first five terms of the sequence in which a₁ =7 , and a_n=3a_n-1-12, if n>2

• to find the 2nd term, use the first term a₁

• a₂ = 3(7) - 12

• a₂ = 9

• to find the 3rd term, use the second term a₃ = 3(9) -12

• a₃ = 15

Recursive Example Find the first five terms of the sequence in which a₁ =-2 , and a_n=(-3)a_n-1+4, if n>2

• a₂=(-3)(-2)+4

• a₂=10

• a₃=-26

• a₄=82

• a₅=-242

Multiple Choice

In recursive formulas what does the term a_n-1 represent?

• the previous term

• the term you are looking for

• the first term

• the last term

Multiple Choice

In recursive formulas what does the term a₁ represent?

• the previous term

• the term you are looking for

• the first term

• the last term

Multiple Choice

Write a recursive formula for the sequence 17,13,9,5...

• a₁ = _17_ , a_n=a_n-1 - 4 __

• a₁ = _13 , a_n=a_n-1 - 4_

• a₁ = _-4 , a_n=a_n-1 -17__

• a₁ = _13 , a_n=a_n-1 +4 ____

Multiple Choice

Which sequence can be found using the recursive formula a₁=6, a_n= 4*a_n-1, n>2

• 6,10,14,20...

• 2,6,10,14...

• 6,24,96,384...

• 6,12,18,24,...

Explicit Formulas

• use to find any term (n^th term)

• need to have the first term (a₁)

• geometric (exponential) a_n = a₁ * r^n-1

• arithmetic (linear) a_n = a₁+ d (n-1)

• LOOK LIKE REGULAR LINEAR AND EXPONENTIAL FORMULAS WHEN SIMPLIFIED :)

Is this arithmetic or geometric?

256,128,64,32,...

Does it have a common difference? or common ratio?

Write an explicit equation

Multiple Choice

Use the explicit formula to find the 7th term of the sequence.
$$a_n=22-3\left(n-1\right)$$  

• 4

• 1

• 18

• 114

Multiple Choice

Use the explicit formula to find the 5th term of the sequence.
$$a_n=3\cdot2^{\left(n-1\right)}$$  

• 24

• 7,776

• 48

• 96

Multiple Choice

Determine if the following sequence is arithmetic or geometric.

6, 30, 150, 750, ...

• Arithmetic

• Geometric

Multiple Choice

Determine if the following sequence is arithmetic or geometric.

2, 5, 8, 11, 14, ...

• Arithmetic

• Geometric

Multiple Choice

Write the explicit formula for the sequence.

10, -2, -14, -26, ...

• $$a_n=10\cdot\left(-\frac{1}{5}\right)^{\left(n-1\right)}$$

• $$a_n=-2\cdot7^{\left(n-1\right)}$$

• $$a_n=10-8\left(n-1\right)$$

• $$a_n=10-12\left(n-1\right)$$

Multiple Choice

Find the first four terms of the sequence.

a_n= 3(4)ⁿ

• 12, 48, 192, 768

• 12, 144, 1728, 20736

• 12, 24, 36, 48

• 3, 12, 48, 192