Word problems with geometric sequences

Sequences, grade 12 (4 units) Lesson 17 of 20Learning

Word problems with geometric sequences

Example: viewers of a video

A video was put on the town council’s website on Sunday.
Each day the number of its viewers was a fixed number of times the number on the day before.
On Wednesday the number of viewers was 8 times the number on Sunday.
On Friday there were 480 viewers more than on Thursday.

The numbers of viewers are a geometric sequence; Sunday is $\displaystyle a_1$ , and Wednesday $\displaystyle a_4$ .
$\displaystyle {\frac{a_4}{a_1}}={q^3}={8}$ , so $\displaystyle {q}={2}$ : the number of viewers doubled each day.
$\displaystyle {a_6-a_5}={a_1\cdot2^5-a_1\cdot2^4}={16a_1}={480}$ , so on Sunday there were $\displaystyle {a_1}={30}$ viewers.
In the seven days: $\displaystyle {S_7}={\frac{30(2^7-1)}{2-1}}={3810}$ viewers.

Exercise 1

The council gives every viewer of the video a discount coupon worth 50 ₪.

a

How much did the coupons of the first seven days cost the council?

/ – 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 council’s budget for all the coupons, from the first day, is 766,500 ₪, and the viewers keep doubling every day.
For how many more days, after the first seven, will the budget last?

/ – 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

A water tank fills up in 5 hours.
The ratio between the water flowing into the tank in each hour and in the hour before is $\displaystyle q$ .
In the first two hours 48 m³ flowed in.
In the first four hours twice as much flowed in as in the last four hours.

a

Find $\displaystyle q$ .

\displaystyle q

/ – 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

Find the volume of the tank.

/ – 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 worker’s yearly salary in her first year is 96,000 ₪.
Each year the salary rises by 5%.

a

What will her yearly salary be in the sixth year? Round to whole shekels.

/ – 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

How much will she earn in all in the first 10 years? Round to whole shekels.

/ – 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

Dana sends a message to 3 friends.
Everyone who gets the message sends it to 3 new friends, and so on, in rounds.
In the first round Dana sent it, and in each round only those who got the message in the round before send it.
After how many rounds have 1,092 people in all got the message?

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