Math Problem Statement

Answer each question based on the following arithmetic sequence: 13, 10, 7, 4,...

Solution

Let's break down the problem and solve it step-by-step.

Part (a)

We are given the arithmetic sequence:
13,10,7,4,13, 10, 7, 4, \dots

  1. Part A: Finding a1a_1
    The first term a1a_1 of the sequence is clearly given as 13.
    So,
    a1=13a_1 = 13

  2. Part B: Finding a3a_3
    To find the third term a3a_3, let's first identify the common difference dd.
    From the sequence: d=1013=3d = 10 - 13 = -3

    Now, using the formula for the nn-th term of an arithmetic sequence: an=a1+(n1)da_n = a_1 + (n-1) \cdot d For a3a_3: a3=13+(31)(3)a_3 = 13 + (3-1) \cdot (-3) a3=13+2(3)a_3 = 13 + 2 \cdot (-3) a3=136=7a_3 = 13 - 6 = 7

  3. Part C: Finding the explicit equation
    The general formula for an arithmetic sequence is: an=a1+(n1)da_n = a_1 + (n-1) \cdot d Substituting a1=13a_1 = 13 and d=3d = -3: an=13+(n1)(3)a_n = 13 + (n-1) \cdot (-3) Simplifying: an=133(n1)a_n = 13 - 3(n-1) an=133n+3a_n = 13 - 3n + 3 an=163na_n = 16 - 3n

Thus, the explicit equation is: an=163na_n = 16 - 3n


Part (b)

We are given several choices for the formula of the sequence:

  • Option A: an=13+(n1)(3)a_n = 13 + (n-1)(-3)
    This is correct, as it matches the derived formula from Part (a).

  • Option B: an=3+(n1)(13)a_n = -3 + (n-1)(13)
    This doesn't match the correct form.

  • Option C: an=13+(n+1)(3)a_n = -13 + (n+1)(3)
    This formula is incorrect because it doesn't align with the correct general form of the sequence.

  • Option D: an=3+(n1)(13)a_n = 3 + (n-1)(-13)
    This is also incorrect, as it doesn't match the arithmetic pattern.

So, Option A is the correct answer.


Would you like any more details or have any questions about the solution? Here are 5 questions that expand on this topic:

  1. What is the difference between arithmetic and geometric sequences?
  2. How do you find the sum of the first nn terms in an arithmetic sequence?
  3. Can you derive the formula for the general term of a geometric sequence?
  4. How do you solve for the number of terms in an arithmetic sequence when given the last term?
  5. How does the common difference impact the shape of a sequence?

Tip: In arithmetic sequences, always check for the common difference first. This value controls the entire pattern of the sequence!

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Algebra

Formulas

General formula for an arithmetic sequence: a_n = a_1 + (n-1) * d
Common difference formula: d = a_2 - a_1

Theorems

Arithmetic Sequence Formula

Suitable Grade Level

Grades 7-10