Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

The double-declining balance method is a method of depreciation in which, for each year \(k=1,2\), \(3, \ldots, n\), the value of an asset is decreased by the fraction \(A_{k}=\frac{2}{n}\left(1-\frac{2}{n}\right)^{k-1}\) of its initial cost. (a) If \(n=5\), find \(A_{1}, A_{2}, \ldots, A_{s}\) (b) Show that the sequence in (a) is geometric, and find \(S_{5-}\) (c) If the initial value of an asset is \(\$ 25,000\), how much of its value has been depreciated after 2 years?

Step-by-Step Solution

Answer

The depreciation after 2 years is $16,000.

Step 1: Calculate Fraction for First Year

For the first year, use the formula provided: \( A_{1} = \frac{2}{5}\left(1 - \frac{2}{5}\right)^{1-1} \). Simplifying, we find \( A_{1} = \frac{2}{5} \).

Step 2: Calculate Fraction for Second Year

For the second year, apply the formula: \( A_{2} = \frac{2}{5}\left(1 - \frac{2}{5}\right)^{2-1} \). Simplifying gives \( A_{2} = \frac{2}{5} \cdot \frac{3}{5} = \frac{6}{25} \).

Step 3: Calculate Fraction for Year k

For each subsequent year \( k \), the formula follows as \( A_{k} = \frac{2}{5} \left( \frac{3}{5} \right)^{k-1} \). Calculate explicitly for years 3, 4, and 5: \( A_{3} = \frac{18}{125}, A_{4} = \frac{54}{625}, A_{5} = \frac{162}{3125} \).

Step 4: Verify the Sequence is Geometric

Check the ratio of these terms: \( \frac{A_{2}}{A_{1}} = \frac{6/25}{10/25} = 0.6 \) and \( \frac{A_{3}}{A_{2}} = \frac{18/125}{6/25} = 0.6 \), confirming each term is multiplied by a common ratio, \( r = 0.6 \). Therefore, the sequence is geometric.

Step 5: Find Partial Sum of the Sequence

Use the geometric series sum formula for the first 5 terms: \( S_{5} = A_{1} \frac{1 - r^{5}}{1 - r} \) with \( r = 0.6 \). Substituting in, \( S_{5} = \frac{2}{5} \frac{1 - 0.6^{5}}{1 - 0.6} \). Calculate to get \( S_{5} \approx 0.9936 \).

Step 6: Calculate Depreciation After 2 Years

Calculate the value depreciated after 2 years: \( A_{1} + A_{2} = \frac{2}{5} \times 25000 + \frac{6}{25} \times 25000 \). Simplifying, the total depreciation is \( 10000 + 6000 = 16000 \).

Key Concepts

Depreciation

Depreciation is a crucial concept in accounting, representing the decrease in value of an asset over time. There are various methods of calculating depreciation, with the double-declining balance method being one of the most aggressive. It accelerates the depreciation process, recognizing a larger depreciation expense in the initial years of an asset's life. This can be particularly beneficial for a business's financial statements, allowing for higher deductions early on. To employ this method, a common formula is used:

• For year 1, the depreciation is \( A_1 = \frac{2}{n} \left(1 - \frac{2}{n}\right)^{0} \)
• For each subsequent year \( k \): \( A_k = \frac{2}{n} \left(1 - \frac{2}{n}\right)^{k-1} \)

This method emphasizes the diminishing value of an asset rapidly upfront, which helps in matching the expense with the benefit received from using the asset.

Geometric Sequence

A geometric sequence is a pattern of numbers where each term after the first is obtained by multiplying the previous one by a constant called the common ratio. In the context of the double-declining balance method, the depreciation fractions form a geometric sequence. The sequence begins with \( A_1 = \frac{2}{5} \), and for each subsequent year, the terms \( A_k = \frac{2}{5} \left(\frac{3}{5}\right)^{k-1} \).

• The common ratio \( r \) can be defined as \( r = 0.6 \),
• As verified by calculating \( \frac{A_{k+1}}{A_k} \) for consecutive terms,

which remains consistent at 0.6. This geometric property allows us to easily compute the sum of terms over any period, an essential task for understanding cumulative depreciation over multiple years.

Asset Valuation

Asset valuation is the process of determining the current worth of an asset after accounting for depreciation. When using the double-declining balance method, it ensures that the asset reflects its reduced value more accurately each year. This method poses that more value is lost earlier in an asset's lifecycle, which is often the case in real-world application.For instance, when given an initial value of \(25,000, the calculation after 2 years shows the cumulative depreciation:

• First Year: \( 25000 \times \frac{2}{5} = 10000 \)
• Second Year: \( 25000 \times \frac{6}{25} = 6000 \)

After these two years, the asset's book value is reduced by \)16,000. Accurate asset valuation is fundamental not just for accounting purposes, but also for making informed financial decisions regarding replacements and investments.

Financial Mathematics

Financial mathematics is a vast field that involves the application of mathematical methods to solve problems related to finance. A prime example is the double-declining balance method, which leverages mathematical formulas to compute depreciation. These formulas allow precise calculation of how much an asset depreciates over its lifetime. Additionally, financial mathematics enables businesses to forecast future expenses and asset value. Understanding financial mathematics is key for professionals involved in budgeting, financial planning, and investment analysis, as it provides the tools necessary to predict growth, analyze potential risks, and establish the financial health of an entity. Through these calculations, businesses can create strategies that optimize financial performance and ensure compliance with regulatory requirements.