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Number Sequence

Number Sequence. PREP – I Chapter 6. Number Sequences Consider the following natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, . . . . We are able to continue writing down the numbers because each successive number follows the preceding one according to a specific rule.

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Number Sequence

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  1. Number Sequence PREP – I Chapter 6

  2. Number Sequences Consider the following natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, . . . . We are able to continue writing down the numbers because each successive number follows the preceding one according to a specific rule. For the sequence of natural numbers, the rule is: start with 1, then add 1 to each term to get the next term.

  3. Introducing sequences 4, 8, 12, 16, 20, 24, 28, 32, . . . 1st term 6th term We call a list of numbers in order a sequence. Each number in a sequence is called a term. If terms are next to each other they are referred to as consecutive terms. When we write out sequences, consecutive terms are usually separated by commas.

  4. Naming sequences Here are the examples of some sequences which you may already know: 2, 4, 6, 8, 10, . . . Even Numbers (or multiples of 2) 1, 3, 5, 7, 9, . . . Odd numbers 3, 6, 9, 12, 15, . . . Multiples of 3 5, 10, 15, 20, 25 . . . Multiples of 5 1, 4, 9, 16, 25, . . . Square numbers 1, 8, 27, 64, 125, . . . Cube numbers 1, 2, 4, 8, 16, . . . Powers of 2 (20, 21, 22, 23,…)

  5. Fibonacci-type sequences 13+21 1+1 1+2 2+3 3+5 5+8 8+13 21+34 Can you work out the next three terms in this sequence? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . How did you work these out? This sequence starts 1, 1 and each term is found by adding together the two previous terms. This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

  6. Ascending sequences ×3 +5 +5 +5 +5 +5 +5 +5 ×3 ×3 ×3 ×3 ×3 ×3 When each term in a sequence is bigger than the one before, the sequence is called an ascending sequence. For example, The terms in this ascending sequence increase in equal steps by adding 5 each time. 2, 7, 12, 17, 22, 27, 32, 37, . . . The terms in the ascending sequence may increase by multiplying each term with a common factor such as by 3 here. 3, 6, 9, 12, 15, 18, 21, 24, . . .

  7. Descending sequences ÷ 2 –7 –7 –7 –7 –7 –7 –7 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 When each term in a sequence is smaller than the one before, the sequence is called a descending sequence. For example, The terms in this descending sequence decrease in equal steps by starting at 107 and subtracting 7 each time. 107, 100, 93, 86, 79, 72, 65, 58, . . . The terms in descending sequence may decrease by dividing each term by a common multiple such as 2 in this case. 1024, 512, 256, 128, 64, 32, 16, 8, . . .

  8. Finding the rule +4 +4 +4 +4 +4 +4 +4 We can describe sequences by finding a rule that tells us how the sequence continues. To work out a rule it is often helpful to find the difference between consecutive terms. For example, look at the difference between each term in this sequence: 3, 7, 11, 15 19, 23, 27, 31, . . . This sequence starts with 3 and increases by 4 each time.

  9. Sequences that decrease in equal steps –6 –6 –6 –6 –6 –6 –6 Can you work out the next three terms in this sequence? 22, 16, 10, 4, –2, –8, –14, –20, . . . How did you work these out? This sequence starts with 22 and decreases by 6 each time. Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

  10. Sequences that increase in increasing steps +2 +3 +4 +5 +6 +7 +8 Some sequences increase or decrease in unequal steps. For example, look at the differences between terms in this sequence: 1, 3, 6, 10, 15, 21, 28, 36, . . . This sequence starts with 1 and increases by 2, 3, 4, 5 …

  11. General Term in a Number Sequence The sequence of even numbers 2, 4, 6, 8, 10, 12,… can be rewritten as: 2 x 1, 2 x 2, 2 x 3, 2 x 4, 2 x 5, 2 x 6, … The expression 2 x n or 2n is the formula for the nth term or the general term of the number sequence. By varying the values of the letter n in the formula, we obtain corresponding values of the formula 2n and thus generate terms of the number sequence. The letter n is a variable.

  12. General Term in a Number Sequence The sequence of odd numbers 1, 3, 5, 7, 9, 11,… can be rewritten as: 2 x 1 - 1, 2 x 2 - 1, 2 x 3 - 1, 2 x 4 - 1, 2 x 5 - 1, 2 x 6 - 1, … The formula for the nth term of the sequence of odd numbers is 2n - 1. Similarly, the formula for the nth term of the number sequence, 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36,…. is n2

  13. Sometimes sequences are arranged in a table like this: We can say that each term can be found by multiplying the position of the term by 3. For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence. What is the 100th term in this sequence? 3 × 100 = 300

  14. Consider a pattern: 2 = 1 x 2 6 = 2 x 3 12 = 3 x 4 20 = 4 x 5 . . . 110 = k x (k + 1) Write down The 8th line in the pattern the 8th line is: 72 = 8 x 9 (b) The value of k since 110 = 10 x 11 = 10 x (10 +1), therefore, k = 10

  15. Consider a pattern: 2 + 12 = 3 2 + 22 = 6 2 + 32 = 11 2 + 42 = 18 . . . 2 + x2 = 66 Write down the 6th line in the pattern. the 6th line is 2 + 62 = 2 + 36 = 38 (b) Find the value of x. 2 + x2 = 66 x2 = 66 – 2 x2 = 64 x = √ 64 = 8 Ans

  16. PRACTICE QUESTIONS

  17. Look at these number sequences carefully can you guess the next 2 numbers? What about guess the rule? +10 30 40 50 60 70 80 +3 --------------------------------------------------------------------------------------------------------------------- 17 20 23 26 29 32 --------------------------------------------------------------------------------------------------------------------- -7 48 41 34 27 20 13

  18. Can you work out the missing numbers in each of these sequences? +25 50 75 100 125 150 175 --------------------------------------------------------------------------------------------------------------------- +20 30 50 70 90 110 130 --------------------------------------------------------------------------------------------------------------------- -5 196 191 181 176 171 186 --------------------------------------------------------------------------------------------------------------------- -10 306 296 286 276 266 256

  19. Now try these sequences – think carefully and guess the last number! +1, +2, +3 … 1 2 4 7 11 16 --------------------------------------------------------------------------------------------------------------------- x 2 3 6 12 24 96 48 --------------------------------------------------------------------------------------------------------------------- + 1.5 0.5 2 3.5 5 6.5 8 --------------------------------------------------------------------------------------------------------------------- -3 7 4 1 -2 -5 -8

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