1. Formulas

2. Simple Formulas
You might be given a formula and asked to substitute numbers, e.g.
E = mc2
Find E when m= 90 and c = 3,000,000
E = 90 X (3,000,000 X 3,000,000)
E = 90 x 9,000,000,000,000
E= 810,000,000,000,000

3. Making a formula
Charlene has joined a swimming club. She had to pay £25 to join the club. She also pays £1.50 every time she goes swimming. What is the formula (t = total cost and n = number of swims)
T= £25 + £1.50 x n
Or t= n
How much is it for 30 swims?
T= 25 + (1.5 x 30) T=
T= £70

4. Lets try these;
A goods train has an engine 6m long. Each wagon is 8m long. Write down a formula for the total length of the goods train (T= total length, n = number of wagons). Use this formula to find the total length of a train with 20 wagons
Year 9 are having a party. It costs £90 to hire a disco and £3 per pupil for refreshments. Find a formula for the total cost of the party (T = total cost, n = number of pupils). How much would it cost if there are 120 pupils in year 9?

5. Answers
T = 8n + 6
T = 166m
T = 3n + 90
T = £450

6. Formulas with n2
When you are given a sequence of numbers sometimes you can identify patterns e.g.
1,4,9,16,25……
No. 1 2 3 4 5
Sequence 9 16 25
How do you get from 1 to 1, 2 to 4, 3 to 9, 4 to 16 and 5 to 25
You square these numbers 3 x 3 =9

7. Formulas with n2
We must find if there is a pattern in this sequence, we do this by taking away
Sequence 1 4 9 16 25 3 5 7 9 2 2 2
Because we had to find the difference twice this means that the number needs to be squared.
We half the number we end up with, in this case 2 to find out if we multiply this squared number
What is our formula?

8. Formulas with n2 N 1 2 3 4 5 Sequence 9 16 25 n2
Our formula must be 1n2

9. Formulas with 2n etc Is there a pattern in this sequence:
3,5,7,9,11…..
Sequence 3 5 7 9 11
Difference
Because the difference is two you must multiply the number by two,
Number 1 2 3 4 5 2n
These numbers are all one short of our sequence, so our formula must be 2n + 1

10. Finding the nth term You are given this pattern 6,15,28,45,66….
First you want to find the differences in these numbers
,15, , ,
This tells us that we have 2n2 in our formula

11. Finding the nth term
We then check if that gives us the sequence number N 1 2 3 4 5
Sequence 6 15 28 45 66
2n2 8 18 32 50
Rest 7 10 13 16
We need more in our formula, so the next step is to find how much more

12. Finding the nth term N 1 2 3 4 5 Sequence 6 15 28 45 66 2n2 8 18 32 50
Rest 7 10 13 16 3 3 3 3
This means we must also multiply each number (n) by 3
When 3n is added to the 2n2 number we are 1 short e.g. 3x1is 3 + 2= 5, but the sequence number is 6
What is our final formula?
2n2 + 3n +1

13. Lets try these;
Find the formula for the nth term and the 6th term in the sequence:
3,7,13,21,31,..
6,11,18,27,38,…
3,10,21,36,55,…
6,15,28,45,66,…
9,20,37,60,89,…
2,4,7,11,16,….

14. Answers N2 + n + 1 43 N2 + 2n + 3 51 2n2 + n 78 2n2 + 3n + 1 91
124
N2/2 + n/2 + 1 22

15. Trial and Improvement
This is when you try a number to see how close you are to getting the answer
E.G.
Solve x2 + 3x = 82 (to 1 d.p.)
Lets try x = 7
= 70 (this is too small)
Lets try x = 8
= 88 (this is too big, but is closer to our answer)
Lets try 7.6
= 80.6 (this is 1.4 too small)
Lets try x = 7.7
= 82.4 (this is 0.4 too big, but our closest answer to 1d.p.)
Our answer is x = 7.7

16. Lets try these: Solve x to 1 d.p. X2 + x = 79 X2 + 2x = 19

17. Answers
8.4
3.5
7.9
4.9
1.3
3.9