RECURSIVE SEQUENCE

A recursive sequence is a sequence in which each term is generated using one or more previous terms for which the values are already known. 

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

Example 1 : 

a1 = 28, ak = ak-1 - 4

Solution :

a1  =  28

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

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

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

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

Example 2 : 

a1 = 15, ak = ak-1 + 3

Solution :

a1  =  15

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

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

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

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

Example 3 : 

a1 = 32, ak+1 = ak/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, ak = ak-2 + ak-1

Solution :

a0  =  1

a1  =  3

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

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

a4  =  a4-2 + a4-1  =  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 nth term of the sequence as a function of n (Assume n begins with 1). 

Example 5 : 

a1 = 6, ak+1 = ak + 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, 

an  =  2n + 4

Example 6 : 

a1 = 25, ak+1 = ak - 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, 

an  =  30 - 5n

Example 7 : 

Find the 10th, 11th and 12th terms of an arithmetic sequence, if the common difference is 8 and 9th 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  =  a⋅ r  =  5 ⋅ 2  =  10

a3  =  a⋅ r  =  10 ⋅ 2  =  20

a4  =  a⋅ r  =  20 ⋅ 2  =  40

a5  =  a⋅ r  =  40 ⋅ 2  =  80

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