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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 50 Page 319

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

f(2) = 5 f(5) = 23 f(10) = 254

Practice makes perfect

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

f(1)&=4 f(2)&=5 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)=4	4
2	f( 2)=5	5
3	f( 3)=f( 3-1)+f( 3-2) ⇕ f(3)=f(2)+f(1)	f(3)=5+4 ⇕ f(3)=9
4	f( 4)=f( 4-1)+f( 4-2) ⇕ f(4)=f(3)+f(2)	f(4)=9+5 ⇕ f(4)=14
5	f( 5)=f( 5-1)+f( 5-2) ⇕ f(5)=f(4)+f(3)	f(5)=14+9 ⇕ f(5)=23
6	f( 6)=f( 6-1)+f( 6-2) ⇕ f(6)=f(5)+f(4)	f(6)=23+14 ⇕ f(6)=37
7	f( 7)=f( 7-1)+f( 7-2) ⇕ f(7)=f(6)+f(5)	f(7)=37+23 ⇕ f(7)=60
8	f( 8)=f( 8-1)+f( 8-2) ⇕ f(8)=f(7)+f(6)	f(8)=60+37 ⇕ f(8)=97
9	f( 9)=f( 9-1)+f( 9-2) ⇕ f(9)=f(8)+f(7)	f(9)= 97+60 ⇕ f(9)=157
10	f( 10)=f( 10-1)+f( 10-2) ⇕ f(10)=f(9)+f(8)	f(10)=157+97 ⇕ f(10)=254

Therefore, f(2) = 5, f(5) = 23 and f(10) = 254.

Subchapter links

Communicate Your Answer p.313

Monitoring Progress p.315-317

Exercises p.318-320