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Big Ideas Math Integrated I, 2016 View details

6. Recursively Defined Sequences

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Exercise 40 Page 319

Pay close attention to how the consecutive terms are related.

Recursive Rule: $a_{1} = 10, a_{2} = 9, a_{n} = a_{n - 2} - a_{n - 1}$ Next two terms: $- 22, 37$

Practice makes perfect

We want to write a recursive formula for the given sequence.

10, 9, 1, 8, - 7, 15, \dots

To do so we need to analyze how the consecutive terms are related. Let's find the difference between each pair of consecutive terms.

a_{1} - a_{2} a_{2} - a_{3} a_{3} - a_{4} a_{4} - a_{5} ⋮ = = = = 10 - 9 9 - 1 1 - 8 8 - (- 7) ⋮ = = = = 18 - 7 15 ⋮

$\frac{\frac{1}{1}}{\frac{1}{1}} a_{n - 2} - a_{n - 1} = a_{n} \frac{\frac{1}{1}}{\frac{1}{1}}$

We can see above that the difference of the consecutive terms equals the next term. Therefore, to obtain the value of the term in the

n^{th}

position, we need to subtract two previous terms. With this information and knowing that the first term equals

10

and the second term equals

9,

we can write the recursive formula.

a_{1} = 10, a_{2} = 9 and a_{n} = a_{n - 2} - a_{n - 1}

We will write the next

2

terms of a sequence now.

⋮ a_{5} - a_{6} a_{7} - a_{8} = = ⋮ - 7 - 15 15 - (- 22) = = ⋮ - 22 37

Subchapter links

Communicate Your Answer p.313

Monitoring Progress p.315-317

Exercises p.318-320

Exercise 40 Page 319

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