Math Problem Statement

an-d form of a_n=53+(n-1)*3

Solution

The sequence an=53+(n1)3a_n = 53 + (n-1) \cdot 3 is in arithmetic sequence form where:

  • a1=53a_1 = 53 (the first term)
  • d=3d = 3 (the common difference)

To express this in the general form of ana_n, it simplifies as follows:

an=53+(n1)3a_n = 53 + (n-1) \cdot 3 Distribute 33 to (n1)(n-1):

an=53+3n3a_n = 53 + 3n - 3 Simplify the constants:

an=3n+50a_n = 3n + 50

So, the simplified form of the sequence is:

an=3n+50a_n = 3n + 50

Questions for Deeper Exploration:

  1. What is the 10th term of the sequence?
  2. How do you find the sum of the first 20 terms of this sequence?
  3. If the sequence were geometric instead, how would the formula change?
  4. What is the nn-th term if the first term were a1=60a_1 = 60?
  5. Can you find the inverse function of this sequence, i.e., solving for nn in terms of ana_n?

Tip:

In arithmetic sequences, the difference dd determines the "rate of change" between consecutive terms, so always look for patterns in differences to verify your formula.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Linear Algebra Equations

Formulas

General form of an arithmetic sequence: a_n = a_1 + (n-1)d
Simplified linear form: a_n = 3n + 50

Theorems

Properties of Arithmetic Sequences

Suitable Grade Level

Grades 8-10