4.1. Let aj = 0, a2 = 6, and a3=9. For n > 3, an=an–1+an-3. Show that for all n, an is divisible by 3. %D
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
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![### Problem 4.1
**Given:**
- \( a_1 = 0 \)
- \( a_2 = 6 \)
- \( a_3 = 9 \)
For \( n > 3 \):
\[ a_n = a_{n-1} + a_{n-3} \]
**Objective:**
- Show that for all \( n \), \( a_n \) is divisible by 3.
**Solution Approach:**
1. **Base Cases:**
- \( a_1 = 0 \) is divisible by 3.
- \( a_2 = 6 \) is divisible by 3.
- \( a_3 = 9 \) is divisible by 3.
2. **Inductive Step:**
- Assume \( a_k \), \( a_{k+1} \), and \( a_{k+2} \) are all divisible by 3.
- Show that \( a_{k+3} \) is divisible by 3, given the recursive formula \( a_{n} = a_{n-1} + a_{n-3} \).
By following the process and mathematical induction, you can prove that each term in the sequence is divisible by 3.
This problem explores sequences and recursion, providing insight into how initial conditions and recursive relations define the properties of sequences.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5302243e-bfe3-4faa-a729-c9f62eae2691%2F71862bb3-10db-4a59-b7d2-eeaef9ca6ee3%2Ft2ga93tk_processed.png&w=3840&q=75)
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