Finding nth Terms of Sequences Flashcards

All 7 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 $n^2 + 2n$

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 $x = 13$