Math Problem Statement

  1. For each of these sequences find a recurrence relation satisfied by this sequence. (The answers are not unique because there are infinitely many different recurrence relations satisfied by any sequence .) a) a_{n} = 3 b) a_{n} = 2n; a_{n} = 2n + 3 d) a_{n} = 5 ^ n e) a_{n} = n ^ 2 f) a_{n} = n ^ 2 + n g) a_{n} = n + (- 1) ^ n h ) a_{n} = n! )

Solution

Let's find recurrence relations for each of the given sequences. A recurrence relation expresses each term of the sequence as a function of its preceding terms.

a) an=3a_n = 3

This is a constant sequence. Every term of the sequence is 3. Therefore, the simplest recurrence relation is: an=an−1a_n = a_{n-1} with the initial condition a1=3a_1 = 3. Alternatively, we can also express it as: an=3a_n = 3

b) an=2na_n = 2n

This is an arithmetic sequence with a constant difference of 2 between consecutive terms. The recurrence relation is: an=an−1+2a_n = a_{n-1} + 2 with the initial condition a1=2a_1 = 2.

c) an=2n+3a_n = 2n + 3

This is also an arithmetic sequence, but with a constant difference of 2 between consecutive terms. The recurrence relation is: an=an−1+2a_n = a_{n-1} + 2 with the initial condition a1=5a_1 = 5.

d) an=5na_n = 5^n

This is a geometric sequence with a ratio of 5 between consecutive terms. The recurrence relation is: an=5⋅an−1a_n = 5 \cdot a_{n-1} with the initial condition a1=5a_1 = 5.

e) an=n2a_n = n^2

The difference between consecutive terms is increasing by 2 each time. The recurrence relation is: an=an−1+2n−1a_n = a_{n-1} + 2n - 1 with the initial condition a1=1a_1 = 1.

f) an=n2+na_n = n^2 + n

Here, we can break down the sequence as an=n(n+1)a_n = n(n + 1). The recurrence relation can be written as: an=an−1+2na_n = a_{n-1} + 2n with the initial condition a1=2a_1 = 2.

g) an=n+(−1)na_n = n + (-1)^n

This sequence alternates between adding and subtracting 1 to nn. The recurrence relation is: an=an−1+2(−1)na_n = a_{n-1} + 2(-1)^n with the initial condition a1=0a_1 = 0.

h) an=n!a_n = n!

The factorial sequence is defined by the product of all positive integers up to nn. The recurrence relation is: an=n⋅an−1a_n = n \cdot a_{n-1} with the initial condition a1=1a_1 = 1.

Would you like further clarification or additional details on any of these sequences?

Here are 5 related questions to expand on this topic:

  1. How can we derive a recurrence relation for a sequence that is neither arithmetic nor geometric?
  2. Can you explain how initial conditions affect the solution of a recurrence relation?
  3. What are homogeneous and non-homogeneous recurrence relations?
  4. How can recurrence relations be used to model real-world problems?
  5. Can generating functions be used to solve recurrence relations?

Tip: Understanding the pattern of differences between consecutive terms often helps in deriving the recurrence relation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Recurrence Relations
Arithmetic Sequences
Geometric Sequences
Factorial Function
Quadratic Sequences

Formulas

Recurrence Relation Formula: a_n = f(a_{n-1})
Arithmetic Sequence: a_n = a_{n-1} + d
Geometric Sequence: a_n = r * a_{n-1}
Factorial: a_n = n * a_{n-1}
Quadratic Sequence: a_n = a_{n-1} + 2n - 1

Theorems

Recurrence Relation Theorem
Arithmetic Sequence Theorem
Geometric Sequence Theorem

Suitable Grade Level

College Level

Related Recommendation

Recurrence Relations and Sequence Calculations for u_n

Solving Sequences and Recurrence Relations with Recursive Formulas

Solving Recurrence Relations for Geometric and Arithmetic Sequences

Recursive Definitions of Sequences: Solving the Problem with Recursive Formulas

Recursive Formula for Finding the nth Term of a Sequence