#### Solved Examples and Worksheet for Identifying and Using Recursive Formula

Q1The first term in a sequence is 22. Each term after the first term is 2 more than the term before it. Find the recursive formula for the sequence.
A. an = an - 12, where n ≥ 2
B. a1 = 22, an = an - 1 + 2, where n ≥ 2
C. a1 = 22, an = an - 1 - 2, where n ≥ 2
D. an = 2an - 1, where n ≥ 2

Step: 1
a1 = 22
[First term = 22]
Step: 2
a2 = a1+ 2 = 22 + 2 = 24
[Each term is 2 more than term before it.]
Step: 3
a3 = a2 + 2 = 24 + 2 = 26
Step: 4
an = an - 1 + 2
[This is a recursive formula.]
Step: 5
The recursive formula for the sequence is a1 = 22, an = an - 1 + 2, where n ≥ 2.
Correct Answer is :   a1 = 22, an = an - 1 + 2, where n ≥ 2
Q2The numbers of coins of different countries collected by Jason are 7, 11, 15, 19, 23, 27.Find the recursive formula and also find the value of a10

A. an = an + 4, a10 = 43
B. an = an - 1 + 2, a10 = 43
C. an = an - 1 + 4, a10 = 43
D. an = an + 1 + 4, a10 = 43

Step: 1
a1 = 7
[First term]
Step: 2
a2 = 11 = 7 + 4 = a1 + 4
Step: 3
a3 = 15 = 11 + 4 = a2 + 4
Step: 4
a4 = 19 = 15 + 4 = a3 + 4
Step: 5
a5 = 23 = 19 + 4 = a4 + 4
Step: 6
a6 = 27 = 23 + 4 = a5 + 4
Observe the sequence.]
Step: 7
an = an - 1 + 4
Step: 8
The recursive formula is an = an - 1 + 4
Step: 9
a7 = a6 + 4 = 27 + 4 = 31
Step: 10
a8 =a7 + 4 = 31 + 4 = 35
Step: 11
a9 = a8 + 4 = 35 + 4 = 39
Step: 12
a10 = a9 + 4 = 39 + 4 = 43
Step: 13
The 10th term of the sequence = 43.
Correct Answer is :   an = an - 1 + 4, a10 = 43
Q3A survey was conducted by a sports channel, about the favorite sport of the people in a locality. The table shows the favorite sport and the number of people like. Find the recursive process. A. an = an - 1, n ≥ 2
B. a2 = an - 1 + 2, n≥1
C. an = an - 1 + 1, n ≥ 2
D. an = an - 1 + 2, n ≥ 2

Step: 1
a1 = 2
[First term.]
Step: 2
a2 = 4 = 2 + 2 = a 1 + 2
Step: 3
a3 = 6 = 4 + 2 = a2 + 2
Step: 4
a4 = 8 = 6 + 2 = a3 + 3
[Observe the pattern.]
Step: 5
an = an - 1 + 2, n ≥ 2
Step: 6
The recursive process of their scores is an = an - 1 + 2, n ≥ 2.
Correct Answer is :   an = an - 1 + 2, n ≥ 2
Q4The number of roses grown in 4 gardens in a particular month is, 12, 36, 108, and 324. Which of the following is a recursive formula for number of roses grown?

A. an = 3an - 1, n≥2
B. an = 3an + 1 , n≥2
C. an = 3an, n≥2
D. an = 2an - 1, n≥2

Step: 1
a1 = 12
[First term.]
Step: 2
a2 = 36 = 3(12) = 3a1
Step: 3
a3 = 108 = 3(36) = 3a2
Step: 4
a4 = 324 = 3(108) = 3a3
[Observe the pattern.]
Step: 5
an = 3an - 1
Step: 6
The recursive formula is an = 3an - 1, n≥2
Correct Answer is :   an = 3an - 1, n≥2
Q5The 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

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
Q6The 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

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