Math Problem Statement

The table gives values of an arithmetic sequence an for selected values of n. Which of the following linear functions is constructed from the initial value ao (with n = 0) and common difference of the sequence?

Solution

In an arithmetic sequence, each term differs from the previous one by a constant amount called the common difference. This characteristic allows us to express the sequence in terms of a linear function.

To determine the linear function for this arithmetic sequence, let’s denote:

  • The initial value a0a_0 (when n=0n = 0),
  • The common difference dd, which is the amount added to each term to obtain the next term.

The formula for the general term of an arithmetic sequence is: an=a0+nda_n = a_0 + n \cdot d

Here:

  • a0a_0 is the value of the sequence when n=0n = 0,
  • nn is the term number, and
  • dd is the common difference.

Steps to Determine the Linear Function:

  1. Identify a0a_0: Use the value of the sequence at n=0n = 0.
  2. Determine dd: Calculate the difference between consecutive terms.
  3. Construct the Function: Substitute a0a_0 and dd into the formula an=a0+nda_n = a_0 + n \cdot d.

If you have specific values in the table for selected terms ana_n, we can calculate dd and a0a_0 directly.

Would you like to provide the table so I can proceed with constructing the exact function? Or do you have any specific questions on these steps?


Here are five related questions for further understanding:

  1. What if the common difference is negative? How does it affect the sequence?
  2. How can we determine the value of nn for any given term in an arithmetic sequence?
  3. How do we identify if a given sequence is arithmetic just by looking at a table of values?
  4. What if the sequence started from n=1n = 1 instead of n=0n = 0? How would the formula change?
  5. How can we apply arithmetic sequences to solve real-world problems?

Tip: In arithmetic sequences, checking a few consecutive terms for a constant difference quickly reveals whether it is indeed arithmetic.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Linear Functions

Formulas

General term of arithmetic sequence: an = a0 + n * d

Theorems

Properties of Arithmetic Sequences

Suitable Grade Level

Grades 8-10