Math Problem Statement
Write a recursive formula and an explicit formula for the following arithmetic sequence. -2, 6, 14, 22, 30, ...
Solution
For the given arithmetic sequence , let's determine the recursive and explicit formulas.
1. Recursive Formula
In an arithmetic sequence, each term is obtained by adding a common difference to the previous term. To find , we can subtract the first term from the second term:
So, the recursive formula is: where is the -th term, and is the first term of the sequence.
2. Explicit Formula
The explicit formula for an arithmetic sequence is given by:
Substituting and : Simplify the expression:
So, the explicit formula for the sequence is:
Final Formulas:
- Recursive Formula: , with .
- Explicit Formula: .
Would you like further details or explanations on these formulas?
Here are some related questions for further practice:
- How would you find the 10th term using the explicit formula?
- What is the common difference, and why is it essential for an arithmetic sequence?
- How do you convert a recursive formula to an explicit formula?
- Can you identify the type of sequence given only the recursive formula?
- How would the formulas change if the common difference was negative?
Tip: For arithmetic sequences, always identify the common difference first; it simplifies finding both recursive and explicit formulas.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Recursive Formulas
Explicit Formulas
Formulas
Recursive formula: a_n = a_{n-1} + d
Explicit formula: a_n = a_1 + (n - 1) * d
Theorems
Properties of Arithmetic Sequences
Suitable Grade Level
Grades 9-10
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