Math Problem Statement
how do you simplify explicit formulas? use the following sequence as an example: 4, 10, 16, 22, 28,...
Solution
Simplifying explicit formulas involves determining a formula that directly relates the position of a term in a sequence (denoted as ) to its value. Let’s work through the example sequence: .
Steps to Simplify the Explicit Formula:
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Identify the Sequence Type: The difference between consecutive terms is constant: The sequence is arithmetic, with a common difference .
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Write the General Formula for an Arithmetic Sequence: The explicit formula for an arithmetic sequence is: Here:
- (the first term),
- (the common difference).
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Substitute Values:
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Simplify: Expand and simplify the expression:
Final Explicit Formula:
Verifying the Formula:
For : For : For :
This formula correctly generates the sequence.
Would you like further clarification on any step?
Related Questions:
- How do you derive explicit formulas for geometric sequences?
- What are the differences between recursive and explicit formulas?
- Can explicit formulas be used for non-arithmetic sequences?
- How do you find the -th term of a sequence with a variable common difference?
- What is the role of sequences in real-world applications like finance or physics?
Tip:
When simplifying explicit formulas, always verify the formula by substituting a few values of to ensure accuracy.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Explicit Formulas
Linear Functions
Formulas
a_n = a_1 + (n - 1)d
a_n = 6n - 2
Theorems
Properties of Arithmetic Sequences
Suitable Grade Level
Grades 8-10