Practice: a formula by place and a recursion rule

Sequences, grade 12 (4 units) Lesson 3 of 20Practice

Practice: a formula by place and a recursion rule

The sequence: $\displaystyle {a_n}={7n-4}$ .
Calculate $\displaystyle a_{12}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The sequence: $\displaystyle {a_n}={n^2-2n}$ .
Calculate $\displaystyle a_{15}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The sequence: $\displaystyle {a_n}={6n+5}$ .
In which place is the term 131?

The sequence: $\displaystyle {a_n}={2n^2+1}$ .
In which place is the term 99?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A sequence is defined by the recursion rule $\displaystyle {a_1}={-2}$ , $\displaystyle {a_{n+1}}={a_n+5}$ .
Calculate $\displaystyle a_4$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A sequence is defined by the recursion rule $\displaystyle {a_1}={64}$ , $\displaystyle {a_{n+1}}={\frac{a_n}{2}}$ .
Write its first five terms, and a formula by place for the sequence.

The graph shows the first five terms of a sequence.

\displaystyle (1,12)

\displaystyle (2,6)

\displaystyle (3,4)

\displaystyle (4,3)

\displaystyle (5,2.4)

Is the sequence increasing or decreasing?

Write a formula by place for the sequence.

The sequence: $\displaystyle {a_n}={5n-2}$ .

Work out $\displaystyle a_{n+1}-a_n$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Write a recursion rule for the sequence, with its first term.

Formulas: sequences, grade 12 (4 units)

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A sequence and the sum of its first terms

The term from the sum – $\displaystyle S_n$ the sum of the first $\displaystyle n$ terms
$\displaystyle S_n-S_{n-1}$ $\displaystyle =$ $\displaystyle a_n$ $\displaystyle n\geq2$
The first term is the sum of one term
$\displaystyle a_1$ $\displaystyle =$ $\displaystyle S_1$

The arithmetic sequence

The recursion rule – $\displaystyle d$ the common difference
$\displaystyle a_{n+1}=a_n+d$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1+(n-1)d$
The arithmetic mean of the neighbours
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle \frac{a_{n-1}+a_{n+1}}{2}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(a_1+a_n)}{2}$
The sum without the last term
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(2a_1+(n-1)d)}{2}$

The geometric sequence

The recursion rule – $\displaystyle q$ the common ratio
$\displaystyle a_{n+1}=a_n\cdot q$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1\cdot q^{n-1}$
The geometric mean: a term’s square is the product of its neighbours
$\displaystyle a_n^2$ $\displaystyle =$ $\displaystyle a_{n-1}\cdot a_{n+1}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{a_1(q^n-1)}{q-1}$ $\displaystyle q\neq1$

The infinite geometric sequence

The sum of the infinite sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$