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Common Core (Functions)

Common Core for Mathematics

Examples, solutions, videos, and lessons to help High School students learn how to recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

For example, the Fibonacci sequence is defined recursively by

f(0) = f(1) = 1,

f(n+1) = f(n) + f(n-1) for n ≥ 1.

Common Core: HSF-IF.A.3

The following diagrams show Arithmetic Sequences as Linear Functions and Geometric Sequences as Exponential Functions. Scroll down the page for more examples and solutions.

**Sequences and Functions**

**Arithmetic Sequences as Linear Functions**

An arithmetic sequence is a numerical pattern that increases or decreases at a constant rate called the common difference.

The nth term of an arithmetic sequence with first term a_{1} and a common difference d is given by

a_{n} = a_{1} + (n - 1)d, where n is a positive integer.
Example:

Use the sequence: 1, 10, 19, 28, …

a. Write an equation fr the nth term of the arithmetic sequence.

Step 1: Find the common difference.

Step 2: Write an equation.

b. Find the 12th term of the sequence.

c. Graph the first five terms in the sequence.

d. Which term of the sequence is 172?

**Geometric Sequences as Exponential Functions**

A geometric sequence is a numerical pattern in which each term after the first is found by multiplying the previous by a constant r, called the common ratio.

The nth term a_{n} of a geometric sequence with first term a_{1} and common ratio r is given by the following formula, where n is any positive integer and a_{1}, r ≠, 0.

a_{n} = a_{1}r^{n-1}

**Arithmetic and Geometric Sequences**

**Arithmetic Sequences and Functions**

A(n) = A(1) + (n -1)d

**Writing Arithmetic Sequences as Functions**

This video shows you how to view arithmetic sequences as functions, so that you can write a formula that’ll give you any term of a sequence you want, just by plugging in the number of the term.

**Recursive formulas for Arithmetic and Geometric Sequences**

Arithmetic Sequence: A(n) = A(1) + (n -1)d

Geometric Sequence: T(n) = T(n-1) × r

**Recursive Functions Tutorial (With Fibonacci Sequence)**

This shows you how to solve recursive functions, using the example of the classic Fibonacci recursive function.

f(1) = 1

f(2) = 1

f(n) = f(n-2) + f(n-1)

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