Formulas

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

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= 25 + 1.5n How much is it for 30 swims? T= 25 + (1.5 x 30) T= 25 + 45 T= £70

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?

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

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

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?

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

Formulas with 2n etc Is there a pattern in this sequence: 3,5,7,9,11….. Sequence 3 5 7 9 11 Difference 2 2 2 2 Because the difference is two you must multiply the number by two, Number 1 2 3 4 5 2n 2 4 6 8 10 These numbers are all one short of our sequence, so our formula must be 2n + 1

Finding the nth term You are given this pattern 6,15,28,45,66…. First you want to find the differences in these numbers 6 ,15, 28, 45, 66 9 13 17 21 4 4 4 This tells us that we have 2n2 in our formula

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

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

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,….

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

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 49 + 21 = 70 (this is too small) Lets try x = 8 64 + 24 = 88 (this is too big, but is closer to our answer) Lets try 7.6 57.8 + 22.8 = 80.6 (this is 1.4 too small) Lets try x = 7.7 59.3 + 23.1 = 82.4 (this is 0.4 too big, but our closest answer to 1d.p.) Our answer is x = 7.7

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

Answers 8.4 3.5 7.9 4.9 1.3 3.9