Growth and decay problems

Sequences and induction, grade 11 (5 units) Lesson 15 of 23Learning

Growth and decay problems

Example: compound interest

10,000 ₪ are deposited at 3% interest a year, and the interest joins the deposit.

Each year the amount is multiplied by $\displaystyle 1.03$ : after $\displaystyle n$ years there are $\displaystyle 10000\cdot1.03^n$ ₪ – a geometric sequence with $\displaystyle {q}={1.03}$ .
After 10 years: $\displaystyle 10000\cdot1.03^{10}\approx13{,}439$ ₪.
When will the amount double? By trial: $\displaystyle 1.03^{23}\approx1.97$ and $\displaystyle 1.03^{24}\approx2.03$ – after 24 years.

Example: two salary offers

Two offers, both starting at 4,000 ₪ a month:

A – each month the salary rises by 20 ₪: an arithmetic sequence, $\displaystyle {d}={20}$ .
B – each month the salary rises by 0.25%: a geometric sequence, $\displaystyle {q}={1.0025}$ .

In the first year:

The salary in month 12: A – $\displaystyle {4000+11\cdot20}={4220}$ ₪; B – $\displaystyle 4000\cdot1.0025^{11}\approx4111$ ₪.
The year’s total: A – $\displaystyle {\frac{12(8000+220)}{2}}={49{,}320}$ ₪; B – $\displaystyle \frac{4000(1.0025^{12}-1)}{0.0025}\approx48{,}666$ ₪.

At first B’s rise (10 ₪) is smaller than A’s, but it grows from month to month. Over a very long time the geometric offer overtakes – which is better depends on how long you will work.

Example: a bouncing ball

A ball falls from a height of 10 m. After each bounce it rises to 60% of the height before, and falls again.
How far does the ball travel until it stops?

The first fall: 10 m.
After it each bounce is a rise and a fall of the same height: $\displaystyle 6, 3.6, 2.16, \ldots$ – an infinite geometric sequence with $\displaystyle {q}={0.6}$ .
The whole way: $\displaystyle {10+2\cdot\frac{6}{1-0.6}}={10+30}={40}$ m.

Exercise 1

A laboratory has 500 bacteria, and their number doubles every hour.

How many bacteria will there be after 6 hours?

/ – 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

After how many whole hours will their number pass 100,000? Find it by trial.

/ – 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

Exercise 2

By the legend, the inventor of chess asked for one grain of rice on the first square of the board, 2 on the second, 4 on the third, and so on – twice as many on each square.
The board has 64 squares.
Write the number of grains on the whole board as a power of 2 minus a number.

Exercise 3

A car bought for 120,000 ₪ loses 15% of its value every year.
What will it be worth after 5 years? Round to shekels.

/ – 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

Exercise 4

The Cantor set. Take the segment from 0 to 1, and remove its middle third. From each of the two segments left remove the middle third, and so on without end.

$\displaystyle b_n$ is the number of segments left after stage $\displaystyle n$ . Write a recursion rule and a formula by place.

In stage $\displaystyle n$ , $\displaystyle 2^{n-1}$ segments are removed, each of length $\displaystyle \frac{1}{3^n}$ . What is the sum of the lengths removed in all the stages together?

/ – 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

Formulas: sequences and induction, grade 11 (5 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
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle S_n-S_{n-1}$ $\displaystyle n\geq2$
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
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$