Sequences Flashcards

All 13 cards in this deck

In the nth term of a linear sequence, what does the coefficient of $n$ tell you?

The common difference.
e.g. $3.5n + 11.5$ has common difference $3.5$

What are the steps to find the nth term of a linear sequence?
e.g. 15, 18.5, 22, 25.5

1. Find the common difference $d$: $3.5$
2. Coefficient of $n$ is $d$: $3.5n$
3. Add the constant needed for the 1st term: $3.5n + 11.5$

True or false? Writing down the common difference of a linear sequence is enough to answer "work out an expression for the nth term".

False. The common difference alone scores no marks — you must give the full expression, e.g. $3.5n + 11.5$ (or an equivalent such as $15 + (n-1)\times 3.5$).

What is a quadratic sequence?

A sequence with a constant second difference, whose nth term has the form $an^2 + bn + c$ with $a \neq 0$.

In the nth term $an^2 + bn + c$ of a quadratic sequence, how do you get $a$ from the differences?

$a$ = second difference $\div\,2$.
e.g. second difference $6$ gives $a = 3$

What are the steps to find the nth term of a quadratic sequence?
e.g. 3, 8, 15, 24

1. Second difference $\div\,2$ gives $a$: $2 \div 2 = 1$, so $n^2$
2. Subtract $an^2$ from each term: $2, 4, 6, 8$
3. Find the nth term of what is left and add: $2n$, so $n^2 + 2n$

In an nth term rule, what does $n$ stand for and what values can it take?

The position of the term; $n$ is a positive integer $1, 2, 3, \ldots$
e.g. 10th term of $5n - 2$ is $5 \times 10 - 2 = 48$

What are the steps to find the value of the first negative term of a sequence?
e.g. nth term $318 - 9n$

1. Set up the inequality: $318 - 9n < 0$
2. Solve for $n$ and round up to the next integer: $n > 35.3\ldots$, so $n = 36$
3. Substitute that $n$: $318 - 9 \times 36 = -6$

What are the steps to find the limiting value of a sequence as $n \to \infty$?
e.g. nth term $\dfrac{3n + 4}{n}$

1. Split the fraction: $3 + \dfrac{4}{n}$
2. Any term $\dfrac{k}{n} \to 0$ as $n \to \infty$
3. State what is left: limiting value $= 3$

What are the steps to decide whether a given number is a term of a sequence?
e.g. is 100 a term of the sequence with nth term $4n + 2$?

1. Set the nth term equal to the number: $4n + 2 = 100$
2. Solve for $n$: $n = 24.5$
3. It is a term only if $n$ is a positive integer: $24.5$ is not, so 100 is not a term.

True or false? To subtract the sequence with nth term $8n + 6$ from the one with nth term $42 - 3n$, you can write $42 - 3n - 8n + 6$.

False. The whole nth term must be bracketed: $42 - 3n - (8n + 6) = 36 - 11n$.

What are the steps to find the term number where two sequences have equal terms?

e.g. sequences with nth terms $3n + 2$ and $5n - 6$

1. Set the two nth terms equal: $3n + 2 = 5n - 6$
2. Solve for $n$: $2n = 8$, so $n = 4$

What are the steps to find an unknown term in a quadratic sequence?

e.g. find $x$ in the quadratic sequence $1, 5, x, 25$

1. Write the first differences: $4$, $x - 5$, $25 - x$
2. Second differences are equal: $(x - 5) - 4 = (25 - x) - (x - 5)$
3. Solve: $x - 9 = 30 - 2x$, so $x = 13$