This document provides worked solutions for two problems involving mathematical problem solving. The first problem asks whether it is better to take $1 million now or collect money by doubling an amount each day for 21 days. The solution shows the work of discovering the exponential pattern in the daily amounts, deriving the formula a21-1, and calculating the total as $1,048,576 after 21 days. The second problem involves finding values for 'i' and 'c' where their sum equals 20, and the solution outlines using a trial and error process to find the answer.

1. Chapter 9A – Problem Solving
Worked Solutions
Ex. 9A Q.3
My friend and I are trying to work out whether it is better to take a million dollars now or to
collect the money after 21 days by doing the following. Put in a dollar on the first day,
double that dollar the next day, then double the previous day's dollars and continue this
for 21 days. What would you advise? (Use mathematics to explain your answer.)
Step One: Write down all the information that you have.
Day # Amount of Money
1 1
2 2
3 4
4 8
5 16
6 32
7 64
Step Two: Try to identify a pattern that will lead to a formula. You didn’t
need to go any further with the above pattern, as this should be enough
for you to find the formula from.
Day # Amount of Money Pattern
1 1 20 = 1
2 2 21 = 2
3 4 22 = 4
4 8 23 = 8
5 16 24 = 16
6 32 25 = 32
7 64 26 = 64
Step Three: Discovering the pattern and applying a rule/formula: You
might be asking how I got the indices? iI looked at the numbers. In this
case, I just knew. However, if you notice that you can multiply the
previous number by the same number (in this case 2), you know it is
going to be indices.
There is a pattern. The exponents are 1 less than the number day it is.
Therefore we can form a formula.

2. Chapter 9A – Problem Solving
Worked Solutions
Step Four: Test the formula.
an-1 = Amount
i.e. a is 2, n is the day number.
so for 21 days, we substitute into the formula.
a21-1 = a20
As a = 2, the sum is 220
You type this into the calculator to get your answer.
Ans = $1048576
Step Five: Justification of answer
Now you explain your answer to answer the question fully.
You now need to write down what you have found in sentences.
i.e. You will earn/save more money over 21 days by
doubling the amount every day, then you will by keeping
the million dollars at the start.

3. Chapter 9A – Problem Solving
Worked Solutions
9A – Question 4
n c i
1 5 -2
2 10 -4
3 15 -6
4 20 -8
5 25 -10
6 30 -12
7 35 -14
8 40 -16
9 45 -18
10 50 -20
11 55 -22
12 60 -24
13 -26
14 -28
15 -30
16 -32
17
Number
of i +
number of
c = 20
Step 1: Write down all the information.
Step 2: Write down the problem: i.e. number of
I + number of c = 20
Step 3: Trial and error to find the answer.
Step 4: Check to make sure there is not a
better solution.

4. Chapter 9A – Problem Solving
Worked Solutions