10 Sequences and Series Free Online FlashCards

What distinguishes a sequence from a series?

A sequence is an ordered list of numbers, while a series is the sum of terms from a sequence.

What is a recursive formula?

A recursive formula defines a term using one or more earlier terms and must include an initial value.

For $a_n=3n+1$, what is $a_{20}$?

Using the explicit formula, $a_{20}=3(20)+1=61$.

What is the common difference of $12,17,22,27,\ldots$?

The common difference is $d=17-12=5$, which is also the difference between every pair of consecutive terms.

How can two known terms determine an arithmetic sequence's difference?

For two known terms, calculate the common difference with $d=\frac{a_j-a_i}{j-i}$.

What is the common ratio of $3,12,48,192,\ldots$?

The common ratio is $r=12\div3=4$; each term is multiplied by 4.

What is the explicit formula for a geometric sequence?

The explicit formula for a geometric sequence is $a_n=a_1r^{n-1}$.

What do the parts of $\sum_{k=1}^{n}a_k$ represent?

In $\sum_{k=1}^{n}a_k$, $k$ is the index, 1 is the starting value, $n$ is the ending value, and $a_k$ is the general term.

What formula sums the first $n$ terms of an arithmetic sequence?

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

What formula gives a finite geometric-series sum?

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

When does an infinite geometric series converge to a finite sum?

An infinite geometric series has a finite sum only when $|r|<1$.

Find the sum of $12+6+3+\frac{3}{2}+\cdots$.

Here $a_1=12$ and $r=\frac{1}{2}$, so $S_\infty=\frac{12}{1-\frac{1}{2}}=24$.