Problem 65 - Algebra 2

Question

Write a real-life application that can be described by an arithmetic sequence with common difference -5.

Step-by-Step Solution

Answer

A tank with 50 liters of water draining 5 liters every hour forms an arithmetic sequence: \( a_n = 50 - 5(n-1) \).

Step 1: Identify the Problem Context

Start by thinking of real-life situations where there is a consistent decrease in a quantity. Some examples could include water drainage, temperature drop, or money being spent periodically.

Step 2: Establish the Initial Term

Decide on an initial value or quantity. Suppose a tank initially has 50 liters of water.

Step 3: Determine the Common Difference

The common difference here is given as -5, which means each step or time interval, 5 liters will be drained from the tank.

Step 4: Formulate the Arithmetic Sequence

The sequence can be formed by repeatedly subtracting 5 from the initial term. Starting with 50 liters, the next term would be 45 liters, then 40 liters, and so on.

Step 5: Generalize the Sequence

In an arithmetic sequence, the nth term is given by the formula \( a_n = a_1 + (n-1)d \), where \( a_1 = 50 \) and \( d = -5 \). The sequence can then be expressed as \( a_n = 50 + (n-1)(-5) \).

Key Concepts

Real-life applications of sequences

Arithmetic sequences are not just theoretical concepts but find their place in our everyday lives. A common example is interest-free savings, where regular withdrawals are made. Let's break it down with a simple scenario: Imagine you have a jar of coins totaling $50. If you take out $5 every day, the amount of money decreases consistently. This steady decrease forms an arithmetic sequence.

• Real-life situations: water drainage, temperature drop, scheduled payments, etc.
• Real-life application: consistently spending money, such as a budget.

Understanding these sequences helps us model and predict changes in quantities over time. This predictability and structure make arithmetic sequences invaluable in planning and resource management.

Common difference in arithmetic sequences

In an arithmetic sequence, the common difference is a key concept that plays a vital role. It tells us how much each term increases or decreases relative to the previous one. In our example, the common difference is -5, which simply means each subsequent term in our sequence decreases by 5.

• Common difference indicates the fixed amount subtracted from each term.
• Negative indicates a decreasing sequence; positive would indicate an increase.

Recognizing the common difference helps create a sequence and understand its progression. It dictates the direction and pace of change, giving us a mathematical lens to assess alterations in various contexts.

Formulating arithmetic sequences

Formulating an arithmetic sequence involves establishing the first term and using the common difference to find subsequent terms. Start with any initial term, such as the 50 liters of water in a tank example. Then, incorporate the common difference to create a sequence.To find the nth term in the sequence, use the formula \[a_n = a_1 + (n-1)d\]where:

• \(a_n\) is the nth term
• \(a_1\) is the first term
• \(d\) is the common difference (-5 in our case)

This formula allows you to calculate any term in the sequence. Formulating sequences this way provides an efficient method to manage or predict quantities, such as in inventory systems or budgeting approaches. It creates a clear roadmap for understanding how values evolve, making it an essential tool in both academic and practical applications.