Arithmetic Sequence Real Life Problems

SITUATION:

SITUATION:There are 125 passengers in

the first carriage, 150 passengers in the second carriage and 175 passengers

in the third carriage, and so on in an arithmetic sequence.

PROBLEM:What’s the total number of passengers in the

first 7 carriages?

SOLUTION:The sequence is 125, 150, 175 …

Given: a1= 125; a2= 150; a3= 175

Find: S7=?

an = 125+ (n-1)25a7 = 125+(7-

1)25=275

We can use the formula:

Thus, =1400

Carriage 1st 2nd

3rd

… 7th

First 7 carriages

Number of Passengers

125

150

175

… ? Sn

SITUATION:

SITUATION:There are 130 students in grade

one, 210 students in grade two and 290students in grade three in a primary school, and so on in an

arithmetic sequence.

PROBLEM:What’s the total amount of students

In the primary school?(Primary School has 6 grades)


Given: a1= 130; a2= 210; a3= 290Find: S6= ?

an = 130+(n-1)80a6 = 130+(6-1)80=530


Thus, = 1980

Grade 1st 2nd 3rd … 6th Total from 1st to 6th Grade

Number of Students

130

210 290 … ? Sn

SITUATION:

A car travels 300 m the first minute, 420 m the next minute,

540 m the third minute, and so on in an arithmetic

sequence.

PROBLEM:What’s the total distance the car travels in

5 minutes?


Given: a1= 300; a2= 420; a3= 540Find: S5= ?

an = 300+(n-1)120a5 = 300+(5-1)120=780


Thus, = 2700

Minute First Second Third Fourth Fifth 5 minutes in Total

Distance 300 420 540 … ? Sn

PROBLEM:

SITUATION: A writer wrote 890 words on the

first day, 760 words on the second day and 630 words on the third day, and so on in an arithmetic sequence.

PROBLEM:How many words did the writer

write in a week?


Given: a1= 890; a2= 760; a3= 630Find: s7= ?

an = 890-(n-1)130a7 = 890-(7-1)130=110


Thus, =3500

Day 1st 2nd 3rd … 7th Whole Week

Number of Words

890

760 630 … ? Sn

SITUATION:

You visit the Grand Canyon anddrop a penny off the edge of a cliff.

The distance the penny will fall is 16 feet the first second, 48 feet the

next second, 80 feet the third second, and so on in an

arithmetic sequence.

PROBLEM:What is the total distance the object will fall

in 6 seconds?

SOLUTION:Arithmetic sequence: 16, 48, 80, ...

Given: a1= 16; a2= 48; a3= 80Find: S6= ?

The 6th term is 176.

Now, we are ready to find the sum:

Second 1 2 3 4 5 6 Total distance in 6 seconds

Distance 16 48 80 … … 176 .....

SITUATION:The sum of the interior angles

of a triangle is 180º,of a quadrilateral is 360º

and of a pentagon is 540º.

PROBLEM:Assuming this pattern continues,

find the sum of theinterior angles of a dodecagon (12 sides).

SOLUTION:Given: d=180

Find: a10= ?

This sequence is arithmetic and the common difference

is 180. The 12-sided figure will be the 10th term in

this sequence. Find the 10th term.

180 360 540 ... ?

Sides: 3 4 5 ... 12

Term: 1 2 3 ... ?

SITUATION:After knee surgery, your trainer tells you toreturn to your jogging program slowly. He suggests jogging for 12 minutes each day

for the first week.Each week thereafter, he suggests that you

increase that time by 6 minutes per day.

PROBLEM:How many weeks will it be before you are upto jogging 60 minutes per day?

SOLUTION:Given: a1 60; d=6

Find: n= ?

Adding 6 minutes to the weekly jogging time for each week creates the

sequence: 12, 18, 24, ...This sequence is arithmetic.

Week Number

1 2 3 … ?

Minutes of Jogging each day inside the week

12 18 24 … n

SITUATION:20 people live on the first floor of

the building, 34 people on the second floor and 48 people on the

third floor, and soon in an arithmetic sequence.

PROBLEM:What’s the total number of people living in

the building?


Given: a1= 20; a2= 34; a3= 48Find: S5= ?

Floor 1st 2nd 3rd 4th 5th People living in the building

Number of People who live

20 34 48 … ? Sn

an = 20+(n-1)14a5 = 20+(5-1)14=76


Thus, =240

SITUATION:Lee earned $240 in the first week, $350in the second week and $460

in the third week, and so on in an arithmetic

sequence.

PROBLEM:How much did he earn in the first 5 weeks?


Given: a1= 240; a2= 350; a3= 460

Find: S5= ?Week 1st 2nd 3rd 4t

h5th

First 5 weeks

Money that Lee Earned

$240

$350

$460

… ? Sn

an=240+(n-1)110a5=240+(5-1)110=680


Thus, =2300

SITUATION:An auditorium has 20 seats on

the first row, 24 seats on the second row, 28

seats on the third row, and so on and has30 rows of seats

PROBLEM:How many seats are in the theatre?

SOLUTION:Given: a1= 20; a2= 24; a3= 28; n=30

Find: S30= ?

Row 1st 2nd 3rd … 30th Total number of rows

Number of seats

20 24 28 … ? Sn

To find a30 we need the formula for the sequence and then substitute n = 30. The formula for an arithmetic sequence is

We already know that is a1 = 20, n = 30, and the common difference, d, is 4. So now we have

So we now know that there are 136 seats on the 30th row. We can use this back in our formula

for the arithmetic series.

Education

Arithmetic Sequence Real Life Problems