Sequences and how to specify them

Terms of a sequence, finite and infinite sequences, specifying a sequence by listing, by a formula for the nth term, or by a recursive rule, and solving T(n) = k for n.

Terms and notation

A sequence is an ordered list of numbers called terms. We write $T_1$ for the first term, $T_2$ for the second, and $T_n$ for the $n$th term. A sequence is finite if it stops, and infinite if it goes on forever (shown by "$\ldots$" at the end).

All three describe the same sequence. A formula lets you jump straight to any term. A recursive rule only tells you how to get from one term to the next.

Using a formula for the nth term

Example 1●

A sequence has $T_n = n^2 - 3n$. Find the first four terms and $T_{20}$.

$T_1 = 1 - 3 = -2$, $\;T_2 = 4 - 6 = -2$, $\;T_3 = 9 - 9 = 0$, $\;T_4 = 16 - 12 = 4$.

$T_{20} = 400 - 60 = 340$.

Example 2: is it a term?●●

For $T_n = n^2 - 3n$, is $108$ a term? Is $50$ a term?

$n^2 - 3n = 108$ gives $n^2 - 3n - 108 = 0$, so $(n - 12)(n + 9) = 0$.

Set the formula equal to the number and solve for n.

$n = 12$, so $108$ is the $12$th term.

n must be a positive whole number.

$n^2 - 3n - 50 = 0$ has $n = \dfrac{3 \pm \sqrt{209}}{2}$, which is not a whole number. So $50$ is not a term.

Using a recursive rule

Example 3●●

A sequence has $T_1 = 3$ and $T_{n+1} = 2T_n - 1$. Find $T_5$.

$T_2 = 2(3) - 1 = 5$, $\;T_3 = 2(5) - 1 = 9$, $\;T_4 = 17$, $\;T_5 = 33$.

The terms of Example 3

Only the points where n is a whole number are terms. The faint curve just joins them up. Drag A to move from term to term.

Try this

Each term here is $1$ more than a power of $2$. Use this to write a formula for $T_n$.

Flashcards

$T_n$

The $n$th term of a sequence.

Recursive rule

A rule giving each term from the one before, such as $T_{n+1} = 2T_n - 1$.

Is $k$ a term?

Solve $T_n = k$. It is a term only if $n$ is a positive integer.

Practice

For $T_n = n^2 - 3n$, find $T_{20}$.
For $T_n = n^2 - 3n$, which term equals $108$?
If $T_1 = 3$ and $T_{n+1} = 2T_n - 1$, find $T_5$.
Which formula gives the sequence $5, 8, 11, 14, \ldots$?
Which recursive rule gives $5, 8, 11, 14, \ldots$?
For $T_n = 4n - 7$, which term equals $53$?
Is $40$ a term of $T_n = 4n - 7$?
If $T_1 = 2$ and $T_{n+1} = 1 - T_n$, find $T_4$.
For $T_n = (-1)^n + 1$, find $T_7$.