Arithmetic sequence problems

Sequences and induction, grade 11 (5 units) Lesson 8 of 23Learning

Arithmetic sequence problems

Example: a cinema

The first row of a cinema has 20 seats, and each row has 5 seats more than the row before it.
The cinema has 1,125 seats.
How many rows has the cinema?

The numbers of seats in the rows are an arithmetic sequence: $\displaystyle {a_1}={20}$ , $\displaystyle {d}={5}$ .
The seats in $\displaystyle n$ rows: $\displaystyle {\frac{n\left[40+5(n-1)\right]}{2}}={1125}$ .
$\displaystyle {n(5n+35)}={2250}$ , that is $\displaystyle {n^2+7n-450}={0}$ , so $\displaystyle {n}={\frac{-7\pm43}{2}}$ : $\displaystyle {n}={18}$ (the negative solution is rejected).
The cinema has 18 rows, and the last row has $\displaystyle {20+17\cdot5}={105}$ seats.

Example: a salary

A starting salary is 4,000 ₪ a month, and every month it rises by 20 ₪.

The salary in month $\displaystyle n$ : $\displaystyle 4000+20(n-1)$ .
It reaches 5,000 ₪ when $\displaystyle {20(n-1)}={1000}$ , that is $\displaystyle {n}={51}$ : in the 51st month.
In the first year you earn $\displaystyle {\frac{12\left[8000+11\cdot20\right]}{2}}={49{,}320}$ ₪.

Example: two hikers

Two hikers set out on the same morning towards each other, from places 186 km apart.
The first walks 20 km each day.
The second walks 6 km on the first day, and each day 2 km more than the day before.
After how many days do they meet?

After $\displaystyle n$ days the first has walked $\displaystyle 20n$ km, and the second $\displaystyle {\frac{n\left[12+2(n-1)\right]}{2}}={n(n+5)}$ km.
They meet when together they have walked the whole distance: $\displaystyle {20n+n(n+5)}={186}$ , that is $\displaystyle {n^2+25n-186}={0}$ .
$\displaystyle {(n-6)(n+31)}={0}$ , so $\displaystyle {n}={6}$ : they meet at the end of the sixth day.

Exercise 1

The first row of a hall has 16 chairs, and each row has 2 chairs more than the row before it.

a

How many chairs are in the first 30 rows?

/ – 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

b

The hall has 700 chairs. How many rows has it?

/ – 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

Exercise 2

In a flight of stairs each step is 17.5 cm high and 26 cm deep.
The wall under the stairs, from the floor up to the steps, is painted.
The wall under the first step is a rectangle of 26 by 17.5 cm, under the second – 26 by 35 cm, and so on.
What is the area of the wall under 15 steps, in cm²?

/ – 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

Exercise 3

A pile of logs has 3 logs in its top row, and each row has one log more than the row above it.
The pile has 150 logs.
How many rows has the pile, and how many logs are in its bottom row?

Number of rows

/ – 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

Bottom row

/ – 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

Exercise 4

Calculate the sum of all the even numbers from 2 to 100, and the sum of the numbers between 1 and 100 divisible by 3.

a

/ – 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

b

/ – 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

Exercise 5

A starting salary is 6,000 ₪, and every month it rises by 30 ₪.
After how many months will all the salaries add up to 125,700 ₪?

/ – 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

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
$\displaystyle 1^3+2^3+3^3+\ldots+n^3$ $\displaystyle =$ $\displaystyle \frac{n^2(n+1)^2}{4}$