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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 57 Page 320

Practice makes perfect

a

We know that the explicit rule defines an arithmetic sequence. a_n=a_1+(n-1)d When we substitute ( n-1) for n in the explicit rule, we get a_(n-1)=a_1+[(n-1)-1]d. a_n=a_1+(n-1)d ⇓ a_(n-1)=a_1+[( n-1)-1]d

b We will use the equation in Part A when showing the recursive equation. Before that, we need to generate a_1+[(n-1)-1]d on the right hand side of the explicit rule.

a_n=a_1+(n-1)d

Identity Property of Addition

a_n=a_1+[(n-1)+0]d

Rewrite 0 as (- 1)-(- 1)

a_n=a_1+[(n-1)-1+1)]d

AssociativePropAdd

Associative Property of Addition

a_n=a_1+[((n-1)-1)+1]d

Distribute d

a_n=a_1+[(n-1)-1]d+d

a_1+[(n-1)-1]d= a_(n-1)

a_n=a_(n-1)+d

We have showed that the recursive rule for a sequence can be obtained by manipulating the explicit rule.

Subchapter links

Communicate Your Answer p.313

Monitoring Progress p.315-317

Exercises p.318-320