Sequences and the Binomial Theorem Sequences Arithmetic Sequences Geometric Sequences & Series Binomial Theorem

A sequence is a function whose domain is the set of positive integers. Sequences

An important common usage is in binomial coefficient. The binomial coefficient has, unfortunately, three common notations:

Evaluate: 6!

Sequences in which the first (or first few) terms are assigned values and the nth term is defined by a formula that involves one (or more) terms preceding it is a recursively defined sequence.

Summation Notation

An arithmetic sequence is defined as Arithmetic Sequences

Common difference does not depend on n, therefore the sequence is arithmetic.

Theorem: nth Term of an Arithmetic Sequence

The 6th term of an arithmetic sequence is 31. The 19th term is 109. Find the first term and the common difference. Give a recursive formula for the sequence.

Theorem: Sum of n Terms of an Arithmetic Sequence

Find the sum of the first 30 terms of the sequence {7n + 2}. That is, find

A sequence is geometric when the ratio of successive terms is always the same nonzero number. A geometric sequence is defined recursively as Geometric Sequences & Geometric Series

Determine if the following sequence is geometric 3, 15, 75, 375,... The sequence is geometric with a common ratio of 5.

Theorem: nth Term of a Geometric Sequence

Theorem: Sum of First n Terms of a Geometric Sequence

Theorem: Amount of an Annuity If P represents the deposit made in dollars at each payment period for an annuity at i percent interest per payment period, the amount A of the annuity after n payment periods is

Suppose Yola deposits $500 into a Roth IRA every quarter (3 months). What will be the value of the account in 25 years assuming it earns 9% per annum compounded quarterly?

Theorem: Sum of an Infinite Geometric Series

Theorem: Principle of Mathematical Induction Suppose the following two conditions are satisfied with regard to a statement about natural numbers: CONDITION I: The statement is true for the natural number 1. CONDITION II: If the statement is true for some natural number k, it is also true for the next natural number k + 1. Then the statement is true for all natural numbers. Mathematical Induction

CONDITION I: Show true for n = 1 CONDITION II: Assume true for some number k, determine whether true for k + 1.

The Binomial Theorem Definition: Binomial Coefficient Symbol

The Binomial Theorem Let x and a be real numbers. For any positive integer n, we have