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Magazine Chapter 6: Problem 3
Determine the number of the term whose value is 22 in the series \(2 \frac{1}{2}, 4,5 \frac{1}{2}, 7, \ldots\)
Short Answer
Expert verified
The 14th term in the series is 22.
Step by step solution
01
Identify the Pattern
Observe the series provided: \(2 \frac{1}{2}, 4, 5 \frac{1}{2}, 7, \ldots\). Notice that this is an arithmetic sequence where each term increases by a common difference. The first term (\(a\)) is \(2 \frac{1}{2}\), and the difference (\(d\)) can be calculated between two consecutive terms. From \(2 \frac{1}{2}\) to \(4\), the difference \(d = 4 - 2.5 = 1.5\). Confirm that this pattern holds for the next increment (\(5.5 - 4 = 1.5\)).
02
Use the Formula for the nth Term of an Arithmetic Sequence
The formula for finding the nth term of an arithmetic sequence is \(a_n = a + (n-1) \cdot d\). Here, \(a = 2.5\) and \(d = 1.5\). You need to find \(n\) for which \(a_n = 22\).
03
Substitute the Values
Substitute the known values into the formula to solve for \(n\): \[22 = 2.5 + (n - 1) \cdot 1.5\]
04
Simplify the Equation
First isolate the expression containing \(n\): \[22 - 2.5 = (n - 1) \cdot 1.5\]This yields:\[19.5 = (n - 1) \cdot 1.5\]
05
Solve for n
Divide both sides by \(1.5\) to solve for \(n - 1\): \[\frac{19.5}{1.5} = n - 1\]This simplifies to:\[13 = n - 1\]
06
Final Step: Calculate n
Add 1 to both sides of the equation to solve for \(n\): \[n = 13 + 1\]Thus, \(n = 14\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the nth Term
In any arithmetic sequence, each term is determined using a specific position in that sequence, usually referred to as the "nth term." Knowing how to find the nth term is crucial for solving any arithmetic-related problems. The nth term formula for an arithmetic sequence is given by:
- \( a_n = a + (n-1) \cdot d \)
- \( a \) is the first term in the sequence
- \( d \) is the common difference between consecutive terms
- \( n \) is the term position in the sequence
- \( a_n \) is the value of the term at position \( n \)
Finding the Common Difference
The common difference in an arithmetic sequence is the consistent amount by which each term increases from one term to the next. It's a hallmark of arithmetic sequences. To find the common difference:
- Select two consecutive terms from the sequence.
- Subtract the earlier term from the later term.
- From \(2.5\) to \(4\), \(d = 4 - 2.5 = 1.5\)
- Confirm with \(5.5 - 4 = 1.5\)
Steps to Problem Solving
Solving problems related to arithmetic sequences involves systematic steps. Let's break down the method used in this exercise:
- Identify the sequence type: Recognize it as arithmetic by noticing a constant difference between terms.
- Determine the first term \( a \) and common difference \( d \): Use the initial terms to find these values.
- Substitute into the nth term formula: Use \( a_n = a + (n-1) \cdot d \) to set up the equation.
- Solve for the desired term position \( n \): Rearrange and simplify the equation to find \( n \).
Detecting a Sequence Pattern
Understanding and detecting patterns in sequences is crucial for mastering the concept of arithmetic series. Here's how you spot patterns and sequences:
- Observe the numbers given in the sequence.
- Calculate differences between consecutive terms to check consistency.
- If consistent, the pattern reveals an arithmetic sequence.
