What is the difference between the explicit formula and recursive formula for sequences when would you use one over the other​

1. Explicit Formula: 
Defines the \(n\)-th term directly as a function of \(n\). 
Example: For an arithmetic sequence \(a_ n = a_ 1 + ( n- 1) d\).

2. Recursive Formula: 
Defines the \(n\)-th term based on previous term(s). 
Example: For an arithmetic sequence \(a_ n = a_ { n- 1} + d\) with \(a_ 1\) given.

3. When to Use Each: 
Explicit Formula: Use when you need to find a specific term quickly without calculating all previous terms. 
Recursive Formula: Use when the relationship between consecutive terms is important or when building the sequence step-by-step.

Supplemental Knowledge:

Mathematically speaking, sequences can be defined using either an explicit formula or a recursive formula. Understanding their respective differences is vital in solving sequence-related problems efficiently.

1. Explicit Formula: 
- An explicit formula allows you to find any term in the sequence directly without needing to know the previous terms. 
- It is expressed as a function of \(n\), where \(n\) is the position of the term in the sequence. 
- Example: For an arithmetic sequence with the first term \(a_ 1\) and common difference \(d\), the explicit formula is: 
\[a_ n = a_ 1 + ( n- 1) d\]
- Use Case: The explicit formula is particularly useful when you need to find a specific term far down in the sequence without calculating all preceding terms.

2. Recursive Formula: 
- A recursive formula defines each term of the sequence based on one or more previous terms. 
- It requires knowing initial conditions (the first few terms of the sequence). 
- Example: For an arithmetic sequence, the recursive formula is: 
\[a_ { n} = a_ { n- 1} + d, \quad \text { with } a_ 1 = c\]
- Use Case: The recursive formula is useful for defining sequences where each term depends on its predecessors, such as in dynamic programming or when modeling processes that evolve step-by-step.

Knowledge in Action:

Imagine you're tracking your savings over time. If you save 100 dollars every month, you could use: 
1. An explicit formula to calculate your total savings after any number of months directly. 
For example, after 12 months: 

Total Savings = Initial Savings + (Number of Months * Monthly Savings)
2. A recursive approach if you're updating your balance monthly based on last month's balance plus this month's savings. 
For example: 
This Month's Balance = Last Month's Balance + Monthly Savings

Both methods are valuable depending on whether you need quick access to any future value (explicit) or if you're building up values incrementally (recursive).

To further explore and practice working with sequences, check out UpStudy’s Algebra Sequences Calculator! This tool helps you understand both explicit and recursive formulas by providing step-by-step solutions and explanations. Whether you're studying for exams or just curious about mathematical patterns, UpStudy has got you covered.