Solved Examples and Worksheet for Explicit and Recursive Formula in Contextual Problems

Q1The heights (in cm) of 5 students are 100, 110, 120, 130, and 140. Find the recursive formula for the heights of students.
A. an = an - 1 + 1, n≥2
B. an = an + 10, n≥2
C. an = an + 1 + 10, n≥2
D. an = an - 1 + 10, n≥2

Solutions
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Step: 1
a1 = 100
  [First time.]
Step: 2
a2 = 110 = 100 + 10 = a1 + 10
Step: 3
a3 = 120 = 110 + 10 = a2 + 10
Step: 4
a4 = 130 = 120 + 10 = a3 + 10
Step: 5
a5 = 140 = 130 + 10 = a4 + 10
  [Observe the pattern.]
Step: 6
an = an - 1 + 10
Step: 7
The recursive formula is an = an - 1 + 10, n≥2
Correct Answer is :   an = an - 1 + 10, n≥2
Q2The scores of four math teams named P, Q, R, and S in an examination are 8, 22, 42, and 68. Which of the following is true for the explicit formula of their scores?

A. an = n2 + n, n≥1
B. an = n2 + 5n, n≥1
C. an = 3n2 + 5n, n≥1
D. an = 3n2 + n, n≥1

Solutions
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Step: 1
a1 = 8 = 3(1)2 + 5(1)
  [First time.]
Step: 2
a2 = 22 = 3(2)2 + 5(2)
Step: 3
a3 = 42 = 3(3)2 + 5(3)
Step: 4
a4 = 68 = 3(4)2 + 5(4)
  [Observe the pattern.]
Step: 5
an = 3n2 + 5n, n≥1
Step: 6
The explicit formula is an = 3n2 + 5n, n≥1
Correct Answer is :   an = 3n2 + 5n, n≥1
Q3The table shows the time taken by Ella to complete her homework on different days. Find an explicit and recursive formulas for the time taken by her.

A. an = 2n - 10, an = an - 1 +2, n>2
B. an = 2n + 10, an = an - 1 +2, n>2
C. an = 2n + 10, an = an - 1 - 2, n>2
D. an = 2n + 10, an = an +2, n>2

Solutions
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Step: 1
a1 = 12 = 2(1) + 10
  [First term.]
Step: 2
a2 = 14 = 2(2) + 10
Step: 3
a3 = 16 = 2(3) + 10
Step: 4
a4 = 18 = 2(4) + 10
Step: 5
a5 = 20 = 2(5) +10
  [Observe the pattern.]
Step: 6
an = 2n + 10
  [Explicit formula.]
Step: 7
An explicit formula ia an = 2n + 10
Step: 8
a1 = 12
  [First time.]
Step: 9
a2 = 14 = 12 + 2 = a1 + 2
Step: 10
a3 = 16 = 14 + 2 = a2 + 2
Step: 11
a4 = 18 = 16 + 2 = a3 + 2
Step: 12
a5 = 20 = 18 + 2 = a4 + 2
  [Observe the pattern.]
Step: 13
an = an - 1 + 2, n>2
  [Recursive formula.]
Step: 14
The recursive formula is an = an - 1 + 2, n >2
Correct Answer is :   an = 2n + 10, an = an - 1 +2, n>2
Q4Cindy counted the number of pencils bought by his 4 friends individually.The number of pencils bought by first 3 frinds are 1, 8, 27 respectively. Find the explicit formula for the sequence and also find the number of pencils bought by his 4th friend.
A. an = 2n2, 32
B. an = n2, 128
C. an = 2n3, 16
D. an = n3, 64

Solutions
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Step: 1
a1 = 1 = 13
  [Pencils bought by 1st friend.]
Step: 2
a2 = 8 = 23
  [Pencils bought by 2nd friend.]
Step: 3
a3 = 27 = 33
  [Pencils bought by 3rd friend.]
Step: 4
an = n3
  [By observing the pattern.]
Step: 5
An explicit formula is an = n3
Step: 6
a4 = 43 = 64
Step: 7
The number of pencils bought by Candy's 4th friend are 64.
Correct Answer is :   an = n3, 64
Q5Jose played 4 matches, Jason played 7 matches, Devin played 10 matches, Charles played 13 matches and Tyler played 16 matches in football tournament. Find an explicit and recursive formulas for this data.
A. an = 3n +1, an = an - 1 +2, n≥2
B. an = 3n +1, an = an - 1 +2, n≥2
C. an = 3n, an = an - 1 +3, n≥2
D. an = 3n +1, an = an - 1 +3, n≥2

Solutions
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Step: 1
Explicit formula
Step: 2
a1 = 4 = 3(1) + 1
  [First term.]
Step: 3
a2 = 7 = 3(2) + 1
Step: 4
a3 = 10 = 3(3) + 1
Step: 5
a4 = 13 = 3(4) + 1
Step: 6
a5 =16 = 3(5) + 1
  [Observe the pattern.]
Step: 7
an = 3n + 1
Step: 8
Recursive formula
Step: 9
a1 = 4
  [First term.]
Step: 10
a2 = 7 = 4 + 3 = a1 + 3
Step: 11
a3 = 10 = 7 + 3 = a2 + 3
Step: 12
a4 = 13 = 10 + 3 = a3 + 3
Step: 13
a5 = 16 = 13 + 3 = a4 + 3
  [Observe the pattern.]
Step: 14
an = an - 1 + 3, n≥2
  [Observe the pattern.]
Step: 15
An explicit formula ia an = 3n + 1
Step: 16
The recursive formula is an = a n - 1 + 3, n≥2
Correct Answer is :   an = 3n +1, an = an - 1 +3, n≥2
Q6The table shows the number of employees of different age groups in a company. Find the explicit formula for the number of employees of different ages and also find the number of employees at the age of 45.


A. an = n, 24
B. an = n, 22
C. an = 4n, 22
D. an = 4n, 24

Solutions
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Step: 1
a1 = 4 = 4(1)
  [Take No of employees at the age of 20 as first term.]
Step: 2
a2 = 8 = 4(2)
Step: 3
a3 = 12 = 4(3)
Step: 4
a4 = 16 = 4(4)
Step: 5
a5 = 20 = 4(5)
  [Observe the pattern.]
Step: 6
an = 4n
Step: 7
The explicit formula is an = 4n
Step: 8
The number of employees at the age of 45 is a6 = 4(6) = 24
  [Substitute the value.]
Correct Answer is :   an = 4n, 24
Q7The students in the 4th grade are going on a field trip to the local paper mill, by 4 buses. The first bus has 15 students, second has 30 students, third has 45 students and fourth bus has 60 students. Which of the following is true for the recursive formula of the number of students?

A. an = an + 1 + 10, n≥2
B. an = an - 1 + 10, n≥2
C. an = an - 1 + 15, n≥2
D. an = an + 1 + 15, n≥2

Solutions
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Step: 1
a1 = 15
  [Take the number of students in the 1st bus as first term.]
Step: 2
a2 = 30 = 15 + 15 = a1 + 15
Step: 3
a3 = 45 = 30 + 15 = a2 + 15
Step: 4
a4 = 60 = 45 + 15 = a3 + 15
  [Observe the sequence.]
Step: 5
an = an - 1 + 15
  [Observe the sequence.]
Step: 6
The recursive formula is an = a n- 1 + 15, n≥2
Correct Answer is :   an = an - 1 + 15, n≥2
Q8Andy counted the number of students in 4 classes that have a dog as pets are 10, 13, 16, and 19. Choose the correct explicit formula for the number of students?

A. an = 3n + 10
B. an = 3n + 7
C. an = 3n + 3
D. an = 10n + 3

Solutions
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Step: 1
a1 = 10 = 3(1) + 7
  [First term.]
Step: 2
a2 = 13 = 3(2) + 7
Step: 3
a3 = 16 = 3(3) + 7
Step: 4
a4 = 19 = 3(4) + 7
  [Observe the pattern.]
Step: 5
an = 3n + 7
Step: 6
The explicit formula is an = 3n + 7
Correct Answer is :   an = 3n + 7
Q9A cafeteria has 4 fruit baskets and each basket has different number of fruits which is in the sequence 5, 11, 21 and 35. Find the explicit formula for the number of fruits.

A. an = 2n2 + 3
B. an = n2 + 3
C. an = 2n2 + 1
D. an = 2n2 + 2

Solutions
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Step: 1
a1 = 5 = 2(1)2+ 3
  [First term.]
Step: 2
a2 = 11 = 2(2)2 + 3
Step: 3
a3 = 21 = 2(3)2 + 3
Step: 4
a4 = 35 = 2(4)2 + 3
  [Observe the pattern.]
Step: 5
an = 2n2 + 3
Step: 6
The explicit formula is an = 2n2 + 3
Correct Answer is :   an = 2n2 + 3
Q10At home over the holiday, Tommy, Mary, Lana and Josh divided their chestnuts which are in the sequence 2, 3, 5, and 9. Which of the following recursive process is a best suit?

A. an = an - 1 + 1, n ≥2
B. an = 2an - 1 + 1, n ≥2
C. an = 2an - 1 - 1, n ≥2
D. an = an - 1 - 1, n ≥2

Solutions
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Step: 1
a1 = 2
  [First term.]
Step: 2
a2 = 3 = 2(2) -1 = 2(a1) - 1
Step: 3
a 3 = 5 = 2(3) - 1 = 2(a2) - 1
Step: 4
a4 = 9 = 2(5) - 1 = 2(a3) - 1
  [Observe the pattern.]
Step: 5
an = 2an - 1 - 1
Step: 6
The recursive formula is an = 2an -1 - 1, n≥ 2
Correct Answer is :   an = 2an - 1 - 1, n ≥2

HighSchool Math