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Big Ideas Math Integrated I, 2016

BI

Big Ideas Math Integrated I, 2016 View details

6. Recursively Defined Sequences

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Exercise 51 Page 319

The given rule means that, after the first term of the sequence, every term f(n) is the difference of the previous terms f(n-1) and f(n-2).

f(2) = 15 f(5) = - 15 f(10) = - 10

Practice makes perfect

We are asked to write the first 2^\text{nd}, 5^\text{th} and 10^\text{th} term of a sequence, given a recursive rule.

f(1)&=10 f(2)&=15 f(n)&=f(n-1)-f(n-2), for n>2 To do so, we will use a table.

n	f(n)=f(n-1)-f(n-2)	f(n)
1	f( 1)=10	10
2	f( 2)=15	15
3	f( 3)=f( 3-1)-f( 3-2) ⇕ f(3)=f(2)-f(1)	f(3)=15-10 ⇕ f(3)=5
4	f( 4)=f( 4-1)-f( 4-2) ⇕ f(4)=f(3)-f(2)	f(4)=5-15 ⇕ f(4)=- 10
5	f( 5)=f( 5-1)-f( 5-2) ⇕ f(5)=f(4)-f(3)	f(5)=- 10-5 ⇕ f(5)=- 15
6	f( 6)=f( 6-1)-f( 6-2) ⇕ f(6)=f(5)-f(4)	f(6)=- 15-(- 10) ⇕ f(6)=- 5
7	f( 7)=f( 7-1)-f( 7-2) ⇕ f(7)=f(6)-f(5)	f(7)=- 5-(- 15) ⇕ f(7)=10
8	f( 8)=f( 8-1)-f( 8-2) ⇕ f(8)=f(7)-f(6)	f(8)=10-(- 5) ⇕ f(8)=15
9	f( 9)=f( 9-1)-f( 9-2) ⇕ f(9)=f(8)-f(7)	f(9)= 15-10 ⇕ f(9)=5
10	f( 10)=f( 10-1)-f( 10-2) ⇕ f(10)=f(9)-f(8)	f(10)=5-15 ⇕ f(10)=- 10

Therefore, f(2) = 15, f(5) = - 15 and f(10) = - 10.

Subchapter links

Communicate Your Answer p.313

Monitoring Progress p.315-317

Exercises p.318-320