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Sat 4 Mar, 2006 07:03 pm
Hi My son has a table for maths which he needs to work out a formula for. He can see the link but can't work out how to write the formula. Maths is not my strong point! Sorry the columns are all squashed to the left. From what he sees column 2 = day x 100 and column 3 = column 2 plus the previous days column 3. I'm sure there's an easy formula!
Day Prize money on each day Total amount
1 $100 $100
2 $200 $300
3 $300 $600
4 $400 $1,000
5 $500 $1,500
6 $600 $2,100
7 $700 $2,800
8 $800 $3,600
9 $900 $4,500
10 $1,000 $5,500
11 $1,100 $6,600
12 $1,200 $7,800
13 $1,300 $9,100
14 $1,400 $10,500
15 $1,500 $12,000
16 $1,600 $13,600
17 $1,700 $15,300
18 $1,800 $17,100
19 $1,900 $19,000
20 $2,000 $21,000
21 $2,100 $23,100
Your son's pretty much got it
Let C1(D) be the value in column 1 for a particular day-D
Then this would be C1(D)=$100*D.
The values for the second column are similarly
C2(D)=C1(D)+the sum of all previous days
C2(D)=$100*D+Σ$100*(D)
The Greek letter Σ in mathematize means sum of
In this case the $100 is a constant and the D is the variable, so the $100 can be taken out of the Σ &
C2(D)=$100*D+$100*Σ(D)= $100*[D+Σ(D)]
Now here's a trick?-a trick told by Euclid in his Elements about 2500 years ago
ΣD=D*(D-1)/2
So one day D and using an more tricks by of the arab philospher Al-Gebra
C2(D)=$100*[D+Σ(D)]=$100[D+D(D-1)/2]=$100
[(2D+D*D-D)/2]=$100[D*D+D]/2
Simplifying slightly
C2(D)=$100*(D+1)*D/2
Now's that the knut of the second column, but it is always worthwhile to check answers
Let's pick a couple
Try Day 10, so D=10
C2(10)=$100*(10+1)*10/2=$100*11*10/2=$100*55=$5500
Golly it seems to work
Let's do another, day 15
D=15
C2(15)=$100*16*15/2=$100*120=$12000
Another correct answer
So formulae for the two columns can be given as
C1(D)=$100*D
C2(D)=$100*(D+1)*D/2
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