Problem 70 - 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 \(n\) is any positive integer, find \(A_{1}, A_{2}, \ldots, A_{n}\). (b) Show that the sequence in (a) is geometric, and find \(S_{n^{-}}\)

Step-by-Step Solution

Answer

The sequence is geometric: \( A_k = \frac{2}{n}(1-\frac{2}{n})^{k-1} \) with sum \( S_{n^{-}} = 1-(1-\frac{2}{n})^n \).

Step 1: Understand the double-declining balance formula

The value of an asset is decreased each year by a fraction \[ A_k = \frac{2}{n}\left(1-\frac{2}{n}\right)^{k-1}\] of its initial cost. Here, \( n \) is the total number of years for depreciation.

Step 2: Calculate the depreciation values for each year

Given the formula \( A_k = \frac{2}{n}\left(1-\frac{2}{n}\right)^{k-1} \), calculate \( A_1, A_2, \ldots, A_n \):- For \( A_1 \):\[ A_1 = \frac{2}{n}\left(1-\frac{2}{n}\right)^{0} = \frac{2}{n} \]- For \( A_2 \):\[ A_2 = \frac{2}{n}\left(1-\frac{2}{n}\right)^{1} = \frac{2}{n}\left(1-\frac{2}{n}\right) \]- Continue this pattern up to \( A_n \), giving:\[ A_k = \frac{2}{n}\left(1-\frac{2}{n}\right)^{k-1} \] for each year \( k \).

Step 3: Identify the sequence as geometric

The sequence \( A_1, A_2, \ldots, A_n \) can be identified as a geometric sequence because each term has a common ratio relative to the previous term.- Common ratio \( r \) is \( 1-\frac{2}{n} \), as each term is obtained by multiplying the previous term by this ratio.

Step 4: Calculate the sum of the geometric sequence up to \( n-1 \)

The sum of the first \( n-1 \) terms of a geometric sequence is given by\[ S_{n^{-}} = A_1 \frac{1-r^{n}}{1-r} \]where \( r = 1-\frac{2}{n} \). Substituting the values:\[ S_{n^{-}} = \frac{2}{n} \frac{1-(1-\frac{2}{n})^{n}}{1-(1-\frac{2}{n})} \]Simplifying the denominator:\( 1-r = \frac{2}{n} \)Therefore:\[ S_{n^{-}} = \frac{2}{n} \frac{1-(1-\frac{2}{n})^{n}}{\frac{2}{n}} = 1-(1-\frac{2}{n})^{n} \]

Step 5: Conclusion

The sequence terms \( A_1, A_2, \ldots, A_n \) are:\[ A_k = \frac{2}{n}\left(1-\frac{2}{n}\right)^{k-1} \] and form a geometric series with a sum up to \( n-1 \) given by:\[ S_{n^{-}} = 1-(1-\frac{2}{n})^{n} \]

Key Concepts

Double-Declining Balance

The double-declining balance method is a form of accelerated depreciation. It decreases an asset's value faster in the earlier years of its useful life. This method effectively allocates more depreciation expense to the initial years and less to the later years. Here's how it works:

• Depreciation is calculated as a fixed fraction of the asset's initial cost, but this fraction decreases each year.
• The formula used is \( A_k = \frac{2}{n} \left( 1-\frac{2}{n} \right)^{k-1} \).

This formula means that each year, a portion of the asset's value is subtracted, starting with the initial cost. As a result, the asset's book value declines rapidly at first, then the rate of decrease slows over time.

Geometric Sequence

In the context of depreciation, the fractions calculated using the double-declining balance method form a geometric sequence. This type of sequence is characterized by each term being a constant multiple of the previous term. For the depreciation formula in question:

• The common ratio \( r \) is \( 1-\frac{2}{n} \), meaning each successive term is derived by multiplying the previous term by this ratio.
• The sequence formed is \( A_1, A_2, \ldots, A_n \), where each term represents a fraction of the initial cost.

Understanding this geometric relationship helps in identifying how the value changes over time in a consistent pattern, facilitating easier computation of total depreciation.

Asset Depreciation

Asset depreciation is a crucial concept in accounting. It helps companies to allocate the cost of a tangible asset over its useful life. This allocation impacts both financial statements and tax computations.

• For the double-declining balance method, assets depreciate quickly at first.
• The diminishing expense each year reflects the asset's decreasing usability or productivity over time.

The goal is to match the depreciation expense with the revenue generated by the asset in each period. This way, the financial statements accurately reflect the cost against the revenue. Using methods like double-declining balance ensures that tax deductions occur prominently in earlier years.

n-Year Depreciation

The term "n-year depreciation" refers to the lifespan over which an asset is depreciated. In accounting, this is the time period during which the asset is expected to be productive or valuable to the business.

• The variable \( n \) in the formula represents this period, generally expressed in years.
• The choice of \( n \) can greatly impact the depreciation calculations and outcomes of each year's financial results.

When using the double-declining balance formula, this time span impacts how steep the depreciation schedule will be in its initial years. A longer \( n \) means a slower decrease in the asset's book value per year, while a shorter \( n \) results in a more aggressive write-off during the asset's earlier years.