Math Problem Statement
Solution
Let's break down the problem and solve it step-by-step.
Part (a)
We are given the arithmetic sequence:
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Part A: Finding
The first term of the sequence is clearly given as 13.
So,
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Part B: Finding
To find the third term , let's first identify the common difference .
From the sequence:Now, using the formula for the -th term of an arithmetic sequence: For :
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Part C: Finding the explicit equation
The general formula for an arithmetic sequence is: Substituting and : Simplifying:
Thus, the explicit equation is:
Part (b)
We are given several choices for the formula of the sequence:
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Option A:
This is correct, as it matches the derived formula from Part (a). -
Option B:
This doesn't match the correct form. -
Option C:
This formula is incorrect because it doesn't align with the correct general form of the sequence. -
Option D:
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:
- What is the difference between arithmetic and geometric sequences?
- How do you find the sum of the first terms in an arithmetic sequence?
- Can you derive the formula for the general term of a geometric sequence?
- How do you solve for the number of terms in an arithmetic sequence when given the last term?
- 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