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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; 1. 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 2. 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 1 4 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 3 9 5 16 7 25 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 1 4 9 16 25 n2 1 4 9 16 25 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 Difference 2 2 2 11 2 Because the difference is two you must multiply the number by two, Number 1 2 2n 2 4 3 6 4 5 8 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 • 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 2 8 18 32 50 Rest 4 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 2 8 18 32 50 Rest 4 7 10 13 16 3 3 3 This means we must also multiply each number (n) by 3 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 3n2 + 2n + 4 124 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: • 1. 2. 3. 4. 5. 6. Solve x to 1 d.p. X2 + x = 79 X2 + 2x = 19 X2 + 4x = 93 39 = x2 + 3x X(7 + x) = 11 X (24 + x) = 110 Answers 1. 2. 3. 4. 5. 6. 8.4 3.5 7.9 4.9 1.3 3.9