Review: sequences

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

Review: sequences

1

A sequence is defined by the recursion rule $\displaystyle {a_1}={2}$ , $\displaystyle {a_{n+1}}={a_n+3n}$ .
Calculate $\displaystyle a_4$ .

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2

The arithmetic sequence: $\displaystyle 17, 13, 9, \ldots, -63$ .

a

How many terms does the sequence have?

b

Calculate the sum of all its terms.

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3

In an arithmetic sequence $\displaystyle {a_5}={14}$ .
The sum of the first 10 terms is 155.
Find $\displaystyle a_1$ and $\displaystyle d$ .

4

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={4n^2-3n}$ .
Find $\displaystyle a_n$ , and prove that the sequence is arithmetic.

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5

A stack of cups has one cup in the top row.
Each row has 2 cups more than the row above it, and the stack has 15 rows.
How many cups are in the stack?

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6

In a geometric sequence $\displaystyle {a_3}={20}$ and $\displaystyle {a_6}={160}$ .

a

Find $\displaystyle q$ .

b

Calculate the sum of the first 8 terms.

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7

In an infinite geometric sequence the sum of the first two terms is 15, and the sum of the sequence is 16.
Find $\displaystyle q$ and $\displaystyle a_1$ .

8

The infinite geometric sequence: $\displaystyle 81, 27, 9, \ldots$
Calculate the sum of the terms in the even places.

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9

In an arithmetic sequence $\displaystyle {a_1}={1}$ and $\displaystyle d\neq0$ .
The terms $\displaystyle a_1$ , $\displaystyle a_3$ , $\displaystyle a_{11}$ (in this order) form a geometric sequence.
Find $\displaystyle d$ and the ratio of the geometric sequence.

\displaystyle d

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\displaystyle q

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10

The sequence: $\displaystyle {a_n}={5\cdot2^{-n}}$ .

a

Work out $\displaystyle \frac{a_{n+1}}{a_n}$ . What does it say about the sequence?

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b

Calculate the sum of all the terms of the sequence.

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Formulas: sequences, grade 12 (4 units)

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
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$