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Three investigations are given below. Use Excel to help you to solve
these problems.
Consecutive Numbers
Some numbers can be expressed as the sum of a string
of consecutive positive numbers (zero is not allowed).
Exactly which numbers have this property?
For example: 9 = 2 + 3 + 4 and 18 = 3 + 4 + 5 + 6
Elevenses
Using numbers which sum to eleven, find the biggest
product which can be made from them.
Square Difference
Which numbers can be expressed as the difference
between 2 perfect squares?
My Rich Aunt
My aunt who is getting on a bit has decided to give me an allowance
each year which will increase annually. She gives me two choices:
Choice 1 £10 the first year, £20 the second year, £30 the third
year, £40 the fourth year and so on.
Choice 2 £2 the first year, £4 the second year, £8 the third year,
£16 the fourth year and so on.
Which choice would you select?
1
2
Think of a number
Add 100 to it
Divide by 99
Record the result
Think of a number
Divide 2 by it
Add 1
Record the result
Use a spreadsheet to explore the problems below.
Investigation 1
You win first prize in the Readers Digest Prize Draw. You are given three options:
Option 1 A tax free lump sum of £250,000
Option 2 £15,000 per year for life
Option 3 A £30,000 tax free lump sum and £12,000 per year for life
Which is the best option?
*Create graphs of your data
*Find the algebraic solutions
Other considerations (i.e. extensions)
*You may decide to invest the lump sum or part of it. Where do you invest? Is
interest simple or compound? Is interest added daily, monthly or annually?
*Do you invest your annual payments?
*What about income tax on the annual payments? Does this added income take
your total income into the higher tax bracket?
Investigation 2
Explore the sequence
xn+1 = (2xn+5)/xn for various xn
What do you notice?
*Graph your generated data.
*Solve the sequence algebraically.
Extension
Explore the sequence
xn+1 = -2A/(A + xn) for a variety of values of A
Justify any findings.
Max Box
You might want to start this investigation by making the box using card.
Start with a square of side 10 cm. If you cut out four identical smaller
squares from each corner, it can be folded into an open-topped box.
10 cm
What size of little square should you cut in order to make the biggest
box? (i.e the box with the largest volume)
Investigate what size of little square should you cut in order to make the
biggest box if you start with a square of side 12 cm.
Try starting with squares of other sizes.
Can you find any relationship between the original size of the side of the
square and the size of the little square to be cut out in order to find the
largest box?