RECURSIVE SEQUENCE

In the following examples 1-4, write the first five terms of the sequence defined recursively.

Example 1 : 

a₁ = 28, a_k = a_k-1 - 4

Solution :

a₁  =  28

a₂  =  a_2-1 - 4  =  a₁ - 4  =  28 - 4  =  24

a₃  =  a_3-1 - 4  =  a₂ - 4  =  24 - 4  =  20

a₄  =  a_4-1 - 4  =  a₃ - 4  =  20 - 4  =  16

a₅  =  a_5-1 - 4  =  a₄ - 4  =  16 - 4  =  12

Example 2 : 

a₁ = 15, a_k = a_k-1 + 3

Solution :

a₁  =  15

a₂  =  a_2-1 + 3  =  a₁ + 3  =  15 + 3  =  18

a₃  =  a_3-1 + 3  =  a₂ + 3  =  18 + 3  =  21

a₄  =  a_4-1 + 3  =  a₃ + 3  =  21 + 3  =  24

a₅  =  a_5-1 + 3  =  a₄ + 3  =  24 + 3  =  27

Example 3 : 

a₁ = 32, a_k+1 = a_k/2

Solution :

a₁  =  32

a₂  =  a₁₊₁  =  a₁/2  =  32/2  =  16

a₃  =  a₂₊₁  =  a₂/2  =  16/2  =  8

a₄  =  a₃₊₁  =  a₃/2  =  8/2  =  4

a₅  =  a₄₊₁  =  a₄/2  =  4/2  =  2

Example 4 : 

a₀ = 1, a₁ = 3, a_k = a_k-2 + a_k-1

Solution :

a₀  =  1

a₁  =  3

a₂  =  a_2-2 + a_2-1  =  a₀ + a₁  =  1 + 3  =  4

a₃  =  a_3-2 + a_3-1  =  a₁ + a₂  =  3 + 4  =  7

a₄  =  a_4-2 + a_4-1  =  a₂ + a₃  =  4 + 7  =  11

In the following examples 5-6, write the first five terms of the sequence defined recursively. Use the pattern to write the n^th term of the sequence as a function of n (Assume n begins with 1). 

Example 5 : 

a₁ = 6, a_k+1 = a_k + 2

Solution :

a₁  =  6

a₂  =  a₁₊₁  =  a₁ + 2  =  6 + 2  =  8

a₃  =  a₂₊₁  =  a₂ + 2  =  8 + 2  =  10

a₄  =  a₃₊₁  =  a₃ + 2  =  10 + 2  =  12

a₅  =  a₄₊₁  =  a₄ + 2  =  12 + 2  =  14

In general, 

a_n  =  2n + 4

Example 6 : 

a₁ = 25, a_k+1 = a_k - 5

Solution :

a₁  =  25

a₂  =  a₁₊₁  =  a₁ - 5  =  25 - 5  =  20

a₃  =  a₂₊₁  =  a₂ - 5  =  20 - 5  =  15

a₄  =  a₃₊₁  =  a₃ - 5  =  15 - 5  =  10

a₅  =  a₄₊₁  =  a₄ - 5  =  10 - 5  =  5

In general, 

a_n  =  30 - 5n

Example 7 : 

Find the 10^th, 11^th and 12^th terms of an arithmetic sequence, if the common difference is 8 and 9^th term is 72. 

Solution :

a₉  =  72  and  d  =  8

a₁₀  =  a₉ + d  =  72 + 8 =  80

a₁₁  =  a₁₀ + d  =  80 + 8 =  88

a₁₂  =  a₁₁ + d  =  88 + 8 =  96

Example 8 : 

Write the first five terms of a geometric sequence whose first term is 5 and common ratio is 2.

Solution :

a₁  =  5  and  r  =  2

a₂  =  a₁ ⋅ r  =  5 ⋅ 2  =  10

a₃  =  a₂ ⋅ r  =  10 ⋅ 2  =  20

a₄  =  a₃ ⋅ r  =  20 ⋅ 2  =  40

a₅  =  a₄ ⋅ r  =  40 ⋅ 2  =  80

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