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 nn) to its value. Let’s work through the example sequence: 4,10,16,22,28,4, 10, 16, 22, 28, \ldots.

Steps to Simplify the Explicit Formula:

  1. Identify the Sequence Type: The difference between consecutive terms is constant: 104=6,1610=6,2216=610 - 4 = 6,\quad 16 - 10 = 6,\quad 22 - 16 = 6 The sequence is arithmetic, with a common difference d=6d = 6.

  2. Write the General Formula for an Arithmetic Sequence: The explicit formula for an arithmetic sequence is: an=a1+(n1)da_n = a_1 + (n - 1)d Here:

    • a1=4a_1 = 4 (the first term),
    • d=6d = 6 (the common difference).
  3. Substitute Values: an=4+(n1)6a_n = 4 + (n - 1) \cdot 6

  4. Simplify: Expand and simplify the expression: an=4+6n6a_n = 4 + 6n - 6 an=6n2a_n = 6n - 2

Final Explicit Formula:

an=6n2a_n = 6n - 2

Verifying the Formula:

For n=1n = 1: a1=6(1)2=4(correct!)a_1 = 6(1) - 2 = 4 \quad \text{(correct!)} For n=2n = 2: a2=6(2)2=10(correct!)a_2 = 6(2) - 2 = 10 \quad \text{(correct!)} For n=3n = 3: a3=6(3)2=16(correct!)a_3 = 6(3) - 2 = 16 \quad \text{(correct!)}

This formula correctly generates the sequence.


Would you like further clarification on any step?

Related Questions:

  1. How do you derive explicit formulas for geometric sequences?
  2. What are the differences between recursive and explicit formulas?
  3. Can explicit formulas be used for non-arithmetic sequences?
  4. How do you find the nn-th term of a sequence with a variable common difference?
  5. 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 nn 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