In the money
'Seconds to make up your mind! - Have fun with this hypothetical problem of sequences'
A long lost relative wants to share their fortune with you! There are 4 different options about how to receive your money in yearly installments. Each one starts with a different amount that changes each year according to a given rule. You are given 2 minutes to make your choice only! Having made your choice, investigate the 4 options to look at their pro's and cons and the eventual wisdom of your choice. Have a bit of fun and learn about different types of sequences and how they grow!
Activity
Here you will find all the details you need about the activity. Instructions for students, downloads etc..... This will look different for different activities.
Aims
The main aim of the activity is to introduce geometric sequences in comparison to arithmetic ones. Using this problem as a starting point the aim is to look at the structure of the two different types of sequences and try to generalise about them. Students will generate terms and calculate sums.
Resources
The problem and tasks are outlined in the In the money activity sheet. There is also a partially completed In the money spread sheet to work with. Teachers can read more in the teacher's notes at the bottom of the page
Description
Here follows an outline of what the task is. If students are not reading this page then the teacher will need to show and give this overview.
- Consider the initial question and make an instinctive choice.
- Using a spreadsheet or otherwise generate the terms of the different sequences for a given period of time and conclude on the wisdom of the decision.
- Make graphs to model the situation.
- Work on the arithmetic sequences and generate some terms.
- Work on the geometric sequences, generate some terms, work on some problems and begin to generalise.
The activity sheet
The following can be downloaded from the resource box above.
Winner !!!
You receive a phone call that says you are the winner of a mystery lottery and that you can choose one of 4 ways in which to be paid your winnings. You have only 30 seconds to decide!
Option 1- $3000 in the 1^{st} year, $3250 in the 2^{nd}, $3500 in the 3^{rd} and so on with the amount received increasing by $250 each year.
Option 2- $500 in the 1^{st} year, $1000 in the 2^{nd}, $1500 in the 3^{rd} and so on with the amount received increasing by $500 each year.
Option 3- $1 in the 1^{st} year, $3 in the 2^{nd}, $9 in the 3^{rd} and so on with the amount received multiplying by 3 each year.
Option 4- $50 in the 1^{st} year, $100 in the 2^{nd}, $200 in the 3^{rd} and so on with the amount received multiplying by 2 each year.
a)– you have 30 seconds to make your choice and record it
b)– Having made your choice, construct an excel spreadsheet that will monitor the amounts received for these four options over a 25 year period. Include a column for each option that shows the ‘total’ or ‘sum’ of money received after 5, 10, 15, 20 and 25 years.
c)– Use the spreadsheet to produce two graphs; one that shows how the annual amount compares with the four options and a second to show how the sum compares.
d)- Discuss the wisdom of your choice!
Identifying Arithmetic Sequences
Once again consider the four options and the sequences that are generated by them (the column that shows annual amount)
- Which two of the four options produce arithmetic sequences?
- For each of these, fill in the table below
Option | ||
U_{1} | ||
d | ||
U_{1} | ||
S_{n} | ||
U_{25} | ||
S_{25} |
Defining Geometric Sequences
Refer now to the other two options (not dealt with above)
- Describe these sequences using words
- Fill in the table below for the other two options
Option | ||
U_{1} | ||
U_{2} | ||
U_{7} | ||
U_{10} |
- Now for both options fill in the table of expressions below
Express | Option | Option |
U_{2} in terms of U_{1} | ||
U_{7} in terms of U_{1} | ||
U_{10} in terms of U_{1} | ||
U_{n }in terms of U_{1} |
Formulae
Use your formula booklet to find the formula for the n^{th} term of a Geometric sequence and compare it to yours
Write down the formula, define the terms and explain what each part means
I did it my way!
As a practising maths teacher I know that most of us like to give activities our own little twist and do them 'our way'. It would be great to add a little collection of 'twists' from users. You can either add your twist to the comments section below or e-mail them directly to me at jamesn@inthinking.co.uk In time some of these twists may appear here....
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