Practice: the geometric sequence

Sequences and induction, grade 11 (5 units) Lesson 16 of 23Practice

Practice: the geometric sequence

1

In a geometric sequence $\displaystyle {a_1}={-3}$ and $\displaystyle {q}={2}$ .
In which place is the term $\displaystyle -384$ ?

2

In a geometric sequence $\displaystyle {a_2}={-6}$ and $\displaystyle {a_5}={48}$ .
Find $\displaystyle a_1$ and $\displaystyle q$ , and calculate $\displaystyle S_6$ .

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3

In a geometric sequence of positive terms $\displaystyle {a_1+a_2}={12}$ and $\displaystyle {a_3+a_4}={48}$ .
Find $\displaystyle a_1$ and $\displaystyle q$ .

\displaystyle a_1

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\displaystyle q

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4

Calculate the sum of the infinite geometric sequence $\displaystyle 40, -10, 2.5, \ldots$

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5

Write $\displaystyle 0.2777\ldots$ as a fraction.

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6

In an infinite geometric sequence the sum is 4 times the first term.
Find $\displaystyle q$ .

7

The infinite geometric sequence: $\displaystyle 64, 32, 16, \ldots$
Calculate the sum of the terms in the odd places.

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8

$\displaystyle a_n$ is an arithmetic sequence with $\displaystyle {a_1}={1}$ and $\displaystyle {d}={2}$ , and $\displaystyle {b_n}={3^{a_n}}$ .
Prove that $\displaystyle b_n$ is a geometric sequence, and find its ratio.

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9

A medicine in the body: each hour 20% of the amount in the body breaks down.
At first there are 400 mg in the body.
After how many whole hours will less than 100 mg be left? Find it by trial.

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Formulas: sequences and induction, grade 11 (5 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
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle S_n-S_{n-1}$ $\displaystyle n\geq2$
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
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n\left[2a_1+(n-1)d\right]}{2}$

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

The laws of powers

A product of powers of one base
$\displaystyle a^m\cdot a^n$ $\displaystyle =$ $\displaystyle a^{m+n}$
A quotient of powers of one base
$\displaystyle \frac{a^m}{a^n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^m\right)^n$ $\displaystyle =$ $\displaystyle a^{mn}$
A power with exponent 0
$\displaystyle a^0$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A power with a negative exponent
$\displaystyle a^{-n}$ $\displaystyle =$ $\displaystyle \frac{1}{a^n}$ $\displaystyle a\neq0$

Sums proved by induction

The sum of the natural numbers
$\displaystyle 1+2+3+\ldots+n$ $\displaystyle =$ $\displaystyle \frac{n(n+1)}{2}$
The sum of the odd numbers
$\displaystyle 1+3+5+\ldots+(2n-1)$ $\displaystyle =$ $\displaystyle n^2$
The sum of the squares
$\displaystyle 1^2+2^2+3^2+\ldots+n^2$ $\displaystyle =$ $\displaystyle \frac{n(n+1)(2n+1)}{6}$
The sum of the cubes
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$