13/14 13. Sequences, Series, and Mathematical Modeling Free Online FlashCards

What is a sequence?

A sequence is an ordered list of numbers. Each number is a term, and its position is represented by its index.

What does an explicit sequence formula provide?

An explicit formula gives a term directly from its position, so a distant term can be found without calculating all preceding terms.

What does a recursive formula require?

A recursive formula defines a term from preceding terms and must include an initial value to start the sequence.

How is an arithmetic sequence identified?

An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference, $d$.

What is the explicit formula for an arithmetic sequence?

The explicit formula is $a_n=a_1+(n-1)d$, where $a_1$ is the first term and $d$ is the common difference.

If $a_4=18$ and $a_{10}=42$, what is $d$?

For an arithmetic sequence, $d=\frac{a_{10}-a_4}{10-4}=\frac{42-18}{6}=4$.

How is a geometric sequence identified?

A geometric sequence has a constant ratio between consecutive nonzero terms. Its explicit formula is $a_n=a_1r^{n-1}$.

Is $4,9,14,19,\ldots$ arithmetic or geometric?

The sequence $4,9,14,19,\ldots$ is arithmetic because its common difference is $5$.

What is the sum formula for an arithmetic series?

The sum of the first $n$ terms is $S_n=\frac{n}{2}(a_1+a_n)$.

What is the finite geometric-series sum formula?

For $r\ne1$, the sum of the first $n$ geometric terms is $S_n=a_1\frac{1-r^n}{1-r}$.

When does an infinite geometric series have a finite sum?

An infinite geometric series converges to a finite sum exactly when $|r|<1$. In that case, $S_\infty=\frac{a_1}{1-r}$.

How is a percentage rate converted to an exponential factor?

For growth at percentage rate $p$, use the growth factor $b=1+p$. For decay, use $b=1-p$, with $p$ written as a decimal.