WRITE RECURSIVE AND EXPLICIT FORMULA FOR THE SEQUENCE
Recursive :
i) For a sequence a1, a2, a3 ... an, a recursive formula is a formula that requires the computation of all previous terms in order to find the value of "an"
ii) We can find the value of a term in the sequence using the value of the previous term.
Explicit :
i) For a sequence a1, a2, a3 ... an, explicit formula is a formula that can compute the value of "an" using is location
ii) We can find the value of a term in the sequence using its position
Write an explicit and recursive formula for the following sequences.
Problem 1 :
-4, -6, -8, -10...
Solution :
-4, -6, -8, -10...
Explicit :
an = a + (n - 1)d
a is the first term.
d is the difference.
d = -6 - (-4)
= -6 + 4
= -2
an = -4 + (n - 1)(-2)
= -4 - 2n + 2
an = -2 - 2n
Recursive :
an = an - 1 + d
an = an - 1 - 2
Problem 2 :
84, 71, 58, 45...
Solution :
84, 71, 58, 45...
Explicit :
an = a + (n - 1)d
a is the first term.
d is the difference.
d = 71 - 84
= -13
an = 84 + (n - 1)(-13)
= 84 - 13n + 13
an = 97 - 13n
Recursive :
an = an - 1 + d
an = an - 1 - 13
Problem 3 :
19, 13, 7, 1...
Solution :
19, 13, 7, 1...
Explicit :
an = a + (n - 1)d
a is the first term.
d is the difference.
d = 13 - 19
= -6
an = 19 + (n - 1)(-6)
= 19 - 6n + 6
an = 25 - 6n
Recursive :
an = an - 1 + d
an = an - 1 - 6
Problem 4 :
9, 17, 25, 33...
Solution :
9, 17, 25, 33...
Explicit :
an = a + (n - 1)d
a is the first term.
d is the difference.
d = 17 - 9
= 8
an = 9 + (n - 1)(8)
= 9 + 8n - 8
an = 1 + 8n
Recursive :
an = an - 1 + d
an = an - 1 + 8
Problem 5 :
-3, -1, 1, 3...
Solution :
-3, -1, 1, 3...
Explicit :
an = a + (n - 1)d
a is the first term.
d is the difference.
d = -1 - (-3)
= 2
an = -3 + (n - 1)(2)
= -3 + 2n - 2
an = -5 + 2n
Recursive :
an = an - 1 + d
an = an - 1 + 2
Problem 6 :
110, 88, 66, 44...
Solution :
110, 88, 66, 44...
Explicit :
an = a + (n - 1)d
a is the first term.
d is the difference.
d = 88 - 110
= -22
an = 110 + (n - 1)(-22)
= 110 - 22n + 22
an = 132 - 22n
Recursive :
an = an - 1 + d
an = an - 1 - 22
Find the next three terms of each arithmetic sequence. Then write its recursive formula.
Problem 7 :
3, 10, 17, 24, ...
Solution :
Common difference :
d = a2 - a1
= 10 - 3
= 7
a5 = a4 + d
= 24 + 7
= 31
a6 = a5 + d
= 31 + 7
= 38
a7 = a6 + d
= 38 + 7
= 45
Hence, the next three terms of the above sequences are 31, 38, and 45.
Recursive :
an = an - 1 + d
an = an - 1 + 7
Problem 8 :
11, 8, 5, 2, ...
Solution :
Common difference :
d = a2 - a1
= 8 - 11
= -3
a5 = a4 + d
= 2 - 3
= -1
a6 = a5 + d
= -1 - 3
= -4
a7 = a6 + d
= -4 - 3
= -7
Hence, the next three terms of the above sequences are -1, -4, and -7.
Recursive :
an = an - 1 + d
an = an - 1 - 3
Problem 9 :
31, 19, 7, -5, ...
Solution :
Common difference :
d = a2 - a1
= 19 - 31
= -12
a5 = a4 + d
= -5 - 12
= -17
a6 = a5 + d
= -17 - 12
= -29
a7 = a6 + d
= -29 - 12
= -41
Hence, the next three terms of the above sequences are -17, -29, and -41.
Recursive :
an = an - 1 + d
an = an - 1 - 12
Problem 10 :
-17, -11, -5, 1, ...
Solution :
Common difference :
d = a2 - a1
= -11 - (-17)
= -11 + 17
= 6
a5 = a4 + d
= 1 + 6
= 7
a6 = a5 + d
= 7 + 6
= 13
a7 = a6 + d
= 13 + 6
= 19
Hence, the next three terms of the above sequences are 7, 13, and 19.
Recursive :
an = an - 1 + d
an = an - 1 + 6
Find the first four terms of each arithmetic sequence.
Problem 11 :
f(n) = f(n - 1) + 5
f(1) = -2
Solution :
f(n) = f(n - 1) + 5
f(2) = f(2 - 1) + 5
f(2) = f(1) + 5
= -2 + 5
f(2) = 3
f(3) = f(3 - 1) + 5
f(3) = f(2) + 5
= 3 + 5
f(3) = 8
f(4) = f(4 - 1) + 5
f(4) = f(3) + 5
= 8 + 5
f(4) = 13
-2, 3, 8, 13,...
Common difference :
d = a2 - a1
= 3 - (-2)
= 3 + 2
= 5
a5 = a4 + d
= 13 + 5
= 18
a6 = a5 + d
= 18 + 5
= 23
a7 = a6 + d
= 23 + 5
= 28
a8 = a7 + d
= 28 + 5
= 33
Hence, the next four terms of the above sequences are 18, 23, 28 and 33.
Problem 12 :
f(n) = f(n - 1) - 1.5
f(1) = 10
Solution :
f(n) = f(n - 1) - 1.5
f(2) = f(2 - 1) - 1.5
f(2) = f(1) - 1.5
= 10 - 1.5
f(2) = 8.5
f(3) = f(3 - 1) - 1.5
f(3) = f(2) - 1. 5
= 8.5 - 1. 5
f(3) = 7
f(4) = f(4 - 1) - 1. 5
f(4) = f(3) - 1. 5
= 7 - 1.5
f(4) = 5.5
10, 8.5, 7, 5.5,...
Common difference :
d = a2 - a1
= 8.5 - 10
= -1.5
a5 = a4 + d
= 5.5 - 1.5
= 4
a6 = a5 + d
= 4 - 1.5
= 2.5
a7 = a6 + d
= 2.5 - 1.5
= 1
a8 = a7 + d
= 1 - 1.5
= -0.5
Hence, the next four terms of the above sequences are 4, 2.5, 1 and -0.5.
Problem 13 :
f(n) = f(n - 1) - 11
f(1) = 27
Solution :
f(n) = f(n - 1) - 11
f(2) = f(2 - 1) - 11
f(2) = f(1) - 11
= 27 - 11
f(2) = 16
f(3) = f(3 - 1) - 11
f(3) = f(2) - 11
= 16 - 11
f(3) = 5
f(4) = f(4 - 1) - 11
f(4) = f(3) - 11
= 5 - 11
f(4) = -6
27, 16, 5, -6,...
Common difference :
d = a2 - a1
= 16 - 27
= -11
a5 = a4 + d
= -6 - 11
= -17
a6 = a5 + d
= -17 - 11
= -28
a7 = a6 + d
= -28 - 11
= -39
a8 = a7 + d
= -39 - 11
= -50
Hence, the next four terms of the above sequences are -17, -28, -39 and -50.
Problem 14 :
f(n) = f(n - 1) + 1/2
f(1) = 3
Solution :
f(n) = f(n - 1) + 1/2
f(2) = f(2 - 1) + 1/2
f(2) = f(1) + 1/2
= 3 + 1/2
f(2) = 7/2
f(3) = f(3 - 1) + 1/2
f(3) = f(2) + 1/2
= 7/2 + 1/2
= 8/2
f(3) = 4
f(4) = f(4 - 1) + 1/2
f(4) = f(3) + 1/2
= 4 + 1/2
f(4) = 9/2
3, 7/2, 4, 9/2,...
Common difference :
d = a2 - a1
= 7/2 - 3
= (7 - 6)/2
= 1/2
a5 = a4 + d
= 9/2 + 1/2
= 10/2
= 5
a6 = a5 + d
= 5 + 1/2
= 11/2
a7 = a6 + d
= 11/2 + 1/2
= 12/2
= 6
a8 = a7 + d
= 6 + 1/2
= 13/2
Hence, the next four terms of the above sequences are 5, 11/2, 6 and 13/2.
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