WJEC MATHEMATICS INTERMEDIATE ALGEBRA SEQUENCES & Nth TERM 1
Contents Number Machines Continuing a sequence Finding the nth term Writing terms using the nth term Picture Sequences Credits WJEC Question bank http://www.wjec.co.uk/question-bank/question-search.html 2
Number Machines With a number machine, you input a number, the machine changes it, then you get an output. Here's an example, INPUT ADD 5 Multiply 4 OUTPUT So, this machine takes a number, adds 5 to it, then multiplies that number by 4. So what would the output be if the input was 6? Firstly, add 5: 6 + 5 = 11 Then multiply by 4 11 4 = 44 So when the input is 6, the output is 44. You may also be asked to find the input if you are given the output. Key point: When you work backwards through the machine use inverse operations and reverse the order. The above number machine becomes: OUTPUT Divide 4 Subtract 5 INPUT So what would the input be if the output was 36? Firstly, divide by 4: 36 4 = 9 Then subtract 5 9 5 = 4 So when the output is 36, the input is 4. 3
A common question with number machines is 'Find the output when the input is n'. For the previous example we know to firstly add 5, then multiply by 4. Common mistake! n + 5 4 The number machine has to add 5 then multiply by 4. In this common mistake, BIDMAS makes the multiplication happen first. The way to resolve this... BRACKETS! Actual answer! (n + 5 ) 4 It's a very subtle difference that makes a big difference! You may need to do this, but in reverse. Find the input when the output is n Remember to make sure you work through the machine backwards and using the inverse operations. Use brackets if you are unsure! Input = (n 4) 5 4
Exercise A16 1. INPUT ADD 8 Multiply 3 OUTPUT a. Find the output when the input is 7 b. Find the input when the output is 30 c. Find the output when the input is n d. Find the input when the output is n 2. INPUT ADD 3 Multiply 4 OUTPUT a. Find the output when the input is 3 b. Find the input when the output is 48 c. Find the output when the input is n d. Find the input when the output is n 3. INPUT 3 6 OUTPUT a. Find the output when the input is 21 b. Find the input when the output is 4 c. Find the output when the input is n d. Find the input when the output is n 5
Continuing a sequence To write the next terms of a sequence you need to find the 'gap' between each of the terms that are given to you. Example 1 Write the next three terms of the sequence 5, 8, 11,... +3 +3 So, to continue the sequence, continue adding three to each term 5, 8, 11, 14, 17, 20,... Example 2 Sometimes it won't be a simple adding or subtracting sequence. Consider this example. 3, 6, 12,... It's clear you cannot just add or subtract something from each term. So we need to look for something we can multiply or divide each term by. 3, 6, 12, 24, 48, 96, 2 2 6
Exercise A17 Write the next three terms of each of the following sequences. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 4, 7, 10,... 9, 7, 5,... 14, 23, 32,... 2, 6, 18,... 48, 24, 12,... 16, 14, 12,... 13, 17, 21,... 5, 10, 20,... 3, 6, 18,... 8, 4, 2,... If the terms alternate between positive and negative, you will be multiplying or dividing by a negative number 7
Finding the nth term To find the nth term of a sequence, we need two important pieces of information. We need to know the 'gap and we need to know what term would be before the first term. Then use this formula. [gap] n ± [term before first term] Example 3 Find the nth term of the following sequence 7, 10, 13, 16,... The gap here is 3 The term that would be before the 7 is 4 So, the nth term is: 3n + 4 The gap The term before the first term Example 4 Find the nth term of the following sequence 6, 8, 10,... The gap here is 2 The term that would be before the -6 is 4 So, the nth term is: 2n 4 The gap The term before the first term 8
Exercise A18 Write the next three terms of each of the following sequences. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 4, 7, 10,... 6, 14, 22,... 7, 14, 21,... 4, 9, 14,... 10, 19, 28,... 7, 16, 25,... 14, 11, 8,... 14, 23, 32,... 16, 14, 12,... 13, 17, 21,... 9
Writing terms using the nth term Make sure you understand substitution for this section. If you are unsure, use the 'Substitution' booklet. Sometimes, you will be given the nth term (or work it out in a previous part of the question) and you will need to find specific terms. Example 5 Find the first 3 terms and 20th term of the sequence with nth term 4n 2 To find the first three terms of this sequence, we need to substitute n as 1, then 2, then 3. Remember n stands for term number. Term 1: 4 1 2 = 2 Term 2: 4 2 2 = 6 Term 3: 4 3 2 = 10 To find the 20th term, we need to substitute n as 20 Term 20: 4 20 2 = 78 Example 5 Find the first 3 terms and 20th term of the sequence with nth term 3n 2 + 3 Term 1: 3 1 2 + 3 = 3 1 + 3 = 3 + 3 = 9 Term 2: 3 2 2 + 3 = 3 4 + 3 = 12 + 3 = 15 Term 3: 3 3 2 + 3 = 3 9 + 3 = 27 + 3 = 30 Term 20: 3 20 2 + 3 = 3 400 + 3 = 1203 10
Exercise A19 Write the first three terms and the 20th term of the series with each of the following nth terms: 1. 2. 3. 4. 5. n + 5 2n + 6 5n 4 n 2 + 4 n 2 5 11
Picture Sequences For some questions you will not be given the sequence, but given a sequence of pictures. Example 6 How many matches will be needed for pattern 7? Pattern 1 Pattern 2 Pattern 3 Despite this seeming more difficult it isn't much more difficult than a regular nth term question. To start, count the number of matches in each picture to create a sequence: 4, 10, 16,... From here. you can calculate the nth term, and then substitute n = 7 to find the number of matches in pattern 7. 12
Exercise A20 Find how many matches are in the 10th term of each of these picture sequences 1. 2. 3. 13
Exam Questions A6 1. 2. 3. 14
4. 5. 15
6. 7. 16