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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 32 Page 319

Can you find the common difference and the first term just by looking at the explicit formula?

$a_{1} = - 14, a_{n} = a_{n - 1} + 6$

Practice makes perfect

The explicit formula of an arithmetic sequence combines the information provided by the two equations of the recursive form into a single equation.

Recursive : Explicit : a_{1} = a_{1}; a_{n} = a_{n - 1} + d a_{n} = a_{1} + (n - 1) d

In these formulas,

d

is the common difference and

a_{1}

is the first term. Looking at the given explicit formula, we cannot identify the common difference

d

nor the value of the first term

a_{1} .

So let's rewrite the formula.

a_{n} = 6 n - 20

Write as a difference

a_{n} = 6 n - 6 - 14

Factor out $6$

a_{n} = 6 (n - 1) - 14

CommutativePropAdd

Commutative Property of Addition

a_{n} = - 14 + 6 (n - 1)

Now we can identify the common difference

d

and the value of the first term

a_{1} .

a_{n} = - 14 + 6 (n - 1)

We can see that

6

is the common difference and the first term is

- 14 .

Now we have enough information to form a recursive formula for this sequence.

a_{1} = a_{1} a_{n} = a_{n - 1} + d \Rightarrow a_{1} = - 14 a_{n} = a_{n - 1} + 6

Subchapter links

Communicate Your Answer p.313

Monitoring Progress p.315-317

Exercises p.318-320