Sequences Flashcards

All 24 cards in this deck

What is a term of a sequence?

Each separate number (or item) in the sequence.
e.g. in $3, 7, 11$ the 2nd term is $7$.

What is a term-to-term rule?

A rule that tells you how to get from one term to the next.
e.g. "multiply the previous term by $2$ then add $1$".

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

A rule that gives any term directly from its position number $n$.
e.g. $n$th term $= 4n + 3$.

How do you find a particular term from an $n$th term rule?

e.g. find the 9th term when the $n$th term is $4n + 3$

Substitute the position number for $n$.
$4 \times 9 + 3 = 39$.

True or false? In the $n$th term rule $4n + 3$, the letter $n$ stands for the value of the term.

False. $n$ stands for the position in the sequence (1st, 2nd, 3rd, …).

True or false? To generate a sequence from a term-to-term rule you must also be told the first term.

True. The same rule gives different sequences from different starting numbers.

What are the steps to continue a sequence given a term-to-term rule?

e.g. $2, 5, 11, 23$ with rule "multiply the previous term by $2$ then add $1$"

1. Take the last term given: $23$.
2. Apply the rule to it: $23 \times 2 = 46$, $46 + 1 = 47$.
3. Write it as the next term: 5th term $= 47$.

How do you describe the rule for continuing an arithmetic sequence?

e.g. $5, 11, 17, 23$

State what is added (or subtracted) each time.
"Add $6$" — so the next term is $29$.

How do you continue a Fibonacci type sequence?

e.g. $3, 3, 6, 9, 15$

Add the two previous terms.
$9 + 15 = 24$ and $15 + 24 = 39$.

True or false? Every sequence has the same difference between one term and the next.

False. Only arithmetic sequences do; others may multiply, or have changing differences.

True or false? "It goes up by $1$ each time" is an acceptable rule for the sequence $1, 2, 4, 7, 11$.

False. The differences go up by $1$ each time ($+1, +2, +3, +4$), not the terms.

What are the first five triangular numbers?

$1, 3, 6, 10, 15$

What are the first five square numbers?

$1, 4, 9, 16, 25$

What are the first five cube numbers?

$1, 8, 27, 64, 125$

What is a Fibonacci type sequence?

A sequence where each term is the sum of the two terms before it.
e.g. $1, 1, 2, 3, 5, 8$.

What is a geometric progression?

A sequence where each term is multiplied by the same number (the common ratio) to get the next.
e.g. $2, 6, 18, 54$ ($\times 3$ each time).

What is an arithmetic sequence?

A sequence with a constant difference — the same number is added (or subtracted) each time.
e.g. $5, 11, 17, 23$.

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

e.g. $5, 11, 17, 23$

1. Find the common difference: $6$, so start with $6n$.
2. Compare $6n$ ($6, 12, 18$) with the terms: each is $1$ less.
3. $n$th term $= 6n - 1$.

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

The common difference. It is negative if the sequence decreases.
e.g. $20, 17, 14$ has $n$th term $23 - 3n$.

How do you decide whether a given number is a term of a sequence?

e.g. is $85$ a term of the sequence with $n$th term $8n + 16$?

Set the $n$th term equal to the number and solve for $n$; it is a term only if $n$ is a positive whole number.
$8n + 16 = 85$ gives $n = 8.625$, so $85$ is not a term.

How do you find an expression in terms of $n$ for the number of tiles in pattern number $n$?

e.g. patterns needing $4, 8, 12, 16$ tiles

Treat the numbers of tiles as a linear sequence and find its $n$th term.
Difference $4$ and no adjustment needed, so $4n$.

True or false? Every term of the sequence with $n$th term $2n + 1$ is odd.

True. $2n$ is always even, and even $+\,1$ is odd.

How do you find the position of a given number in a sequence?

e.g. the $n$th term is $4n + 3$ — which term is $39$?

1. Put the $n$th term equal to the number: $4n + 3 = 39$
2. Solve for $n$: $n = 9$
3. $n$ is the position: $39$ is the 9th term.

In a linear sequence, how do you find the difference between the 5th term and the 7th term without working out both terms?

e.g. the $n$th term is $6n + 31$

Multiply the common difference by the gap in positions: $6 \times 2 = 12$.