Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: Show that the given sequence is arithmetic, and find the common difference. $$ 53,48,43, \ldots, 58-5 n, \ldots $$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of -5.

Step 1: Identify the sequence pattern

The sequence given is \( 53, 48, 43, \ldots, 58-5n, \ldots \). We need to check if the sequence is arithmetic by verifying if the difference between consecutive terms is constant.

Step 2: Calculate the difference between terms

To find the common difference, calculate the difference between the first two terms: \( 48 - 53 = -5 \), and between the second and third terms: \( 43 - 48 = -5 \). The difference is constant.

Step 3: Confirm the pattern with the general formula

The nth term of the sequence is given by \( 58 - 5n \). For an arithmetic sequence, the nth term can be expressed as \( a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference. Here, the common difference \( d \) is \(-5\) which matches our earlier calculation.

Step 4: Verify consistency of the first term

Substitute \( n=1 \) into \( 58-5n \) to get \( 58 - 5 \cdot 1 = 53 \), which matches the first term of the sequence. This consistency confirms the formula is accurate for an arithmetic sequence.

Key Concepts

Common Difference

An arithmetic sequence is a sequence of numbers in which each term, after the first, is obtained by adding a constant difference to the previous term. This constant is known as the 'common difference'. It represents the uniform interval between consecutive numbers in the series.

In our example sequence, the common difference is found by subtracting the previous term from the subsequent term. For example:

• The difference between the first two terms is calculated as: \( 48 - 53 = -5 \).
• Repeating this calculation for the next pair, \( 43 - 48 = -5 \), shows the difference remains consistent.

The common difference here is \(-5\), meaning each term is 5 less than its predecessor.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant, referred to as the common difference. Fundamentally, this type of sequence is linear, as illustrated by the example provided: 53, 48, 43, ...

Given an arithmetic progression, you can visualize it like stepping down a set of equally spaced stairs. If you start at 53, each step drops you down by 5, matching the consistent negative common difference. Recognizing this regular interval helps identify and validate the consistency of an arithmetic sequence.

In a broader context, AP’s enable mathematicians and students to predict future terms or backtrack to previous ones based on known values. In analytical settings, understanding this pattern applies to various disciplines, including economics, computing, and physics.

Sequence Formula

For arithmetic sequences, mathematical formulas allow you to find any term directly without listing all preceding terms. This formula calculates the nth term of an arithmetic sequence as:\[ a_n = a + (n-1)d \]where:

• \(a_n\) is the nth term,
• \(a\) is the first term of the sequence,
• \(d\) is the common difference,
• \(n\) is the term number.

In the given sequence example, the first term \(a\) is 53, and the common difference \(d\) is \(-5\). According to this formula, any term can be found by plugging \(n\) into:\[ a_n = 53 + (n-1)(-5) \]This corresponds to the general nth-term formula provided: \(58 - 5n\). This verification reassures the consistency and reliability of our calculated arithmetic sequence, enabling efficient determination of terms irrespective of ordering.