2)Define the sequence: a, = (n+1)! (n = 1,2,3, ...) recursively. %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The task is to define the sequence \( a_n = (n+1)! \) where \( n = 1, 2, 3, \ldots \) recursively.
In mathematical terms, a recursive definition provides a way to determine each term of the sequence based on the preceding terms. For this sequence:
- Base Case: Start with an initial value. For example, \( a_1 \) can be initially defined as \( 2! = 2 \).
- Recursive Step: For \( n \geq 1 \), define \( a_{n} = (n+1) \times a_{n-1} \).
This means that each term in the sequence is determined by multiplying the previous term by \( n+1 \). Thus, the sequence builds factorials successively starting from \( 2! \).
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