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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?