Sequences Flashcards

All 23 cards in this deck

What is a term-to-term rule for a sequence?

A rule that gives each term from the previous term.
e.g. "first term 3, add 4 each time" gives $3, 7, 11, \dots$

What is a position-to-term rule ($n$th term rule)?

A rule that gives a term directly from its position number $n$.
e.g. $n$th term $=5n+2$, so the 3rd term is $5\times3+2=17$

What are the steps to generate the first three terms from an $n$th term rule?

e.g. $n$th term $=3n-1$

1. Substitute $n=1$: $3(1)-1=2$
2. Substitute $n=2$: $3(2)-1=5$
3. Substitute $n=3$: $3(3)-1=8$
   Sequence: $2, 5, 8$

In sequence notation, what does $u_{n+1}$ mean?

e.g. $u_{n+1}=2u_n+1$ with $u_1=3$

The term after $u_n$, i.e. the next term.
Here $u_2=2\times3+1=7$

True or false? A term-to-term rule on its own is enough to write out a sequence.

False. You also need the first term (or another given term) to start from.

What are the first five triangular numbers, and how is the sequence formed?

$1, 3, 6, 10, 15$ — add one more each time ($+2, +3, +4, \dots$).

What are the first five cube numbers?

$1, 8, 27, 64, 125$

What is a Fibonacci-type sequence?

A sequence in which each term is the sum of the previous two terms.
e.g. $a,\ 2a,\ 3a,\ 5a,\ 8a$

What is a geometric sequence?

A sequence where each term is found by multiplying the previous term by a fixed number, the common ratio.

How do you find the common ratio of a geometric sequence?

e.g. $\sqrt{5},\ 10,\ 20\sqrt{5},\ \dots$

Divide any term by the term before it.
$10\div\sqrt{5}=2\sqrt{5}$

True or false? The common ratio of a geometric sequence must be a whole number.

False. It can be a fraction or a surd, e.g. $r=\tfrac12$ or $r=\sqrt{2}$.

What is an arithmetic (linear) sequence?

A sequence where the same number, the common difference, is added each time.
e.g. $7, 13, 19, 25$ has common difference $6$

What are the steps to find the $n$th term of a linear sequence?

e.g. $1, 5, 9, 13$

1. Find the common difference: $4$, so start with $4n$
2. Compare $4n$ ($4, 8, 12$) with the terms: each is $3$ less
3. $n$th term $=4n-3$

In the $n$th term of a linear sequence, what does the number in front of $n$ tell you?

The common difference of the sequence.
e.g. $6n+1$ has common difference $6$

How do you use an $n$th term rule to find a later term?

e.g. the 100th term of $3n-2$

Substitute the position number for $n$.
$3\times100-2=298$

What are the steps to decide whether a number is a term of a sequence?

e.g. is $-58$ a term of the sequence with $n$th term $8-6n$?

1. Set the $n$th term equal to the number: $8-6n=-58$
2. Solve for $n$: $n=11$
3. $n$ is a positive integer, so yes, $-58$ is the 11th term

True or false? If solving "$n$th term $=k$" gives $n=4.5$, then $k$ is a term of the sequence.

False. $n$ must be a positive whole number, so $k$ is not a term.

What is the general form of the $n$th term of a quadratic sequence?

$an^2+bn+c$, where $a\neq0$.

How do you get the coefficient of $n^2$ from the second difference of a quadratic sequence?

e.g. constant second difference $4$

Halve the second difference: $a=\tfrac{4}{2}=2$, so the $n$th term contains $2n^2$.

What are the steps to find the $n$th term of a quadratic sequence?

e.g. $3, 9, 17, 27$

1. Second difference is $2$, so halve it: the $n^2$ term is $n^2$
2. Subtract $1, 4, 9, 16$ from the terms: $2, 5, 8, 11$, which has $n$th term $3n-1$
3. $n$th term $=n^2+3n-1$

True or false? In a quadratic sequence the first differences are constant.

False. The second differences are constant; the first differences change by a fixed amount.

What are the steps to find unknown coefficients in an $n$th term rule from given terms?

e.g. $n$th term $=an^2+bn$, 2nd term $=-2$, 4th term $=12$

1. Substitute $n=2$: $4a+2b=-2$
2. Substitute $n=4$: $16a+4b=12$
3. Solve the two equations together: $a=2$, $b=-5$

True or false? A linear sequence with $n$th term $8-6n$ has terms that get smaller as $n$ increases.

True. The coefficient of $n$ is negative ($-6$), so each term is $6$ less than the one before.