Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 55-56: Depreciation methods are sometimes used by businesses and individuals to estimate the value of an asset over a life span of \(n\) years. In the sum-of-year's-digits method, for each year \(k=1,2,3, \ldots, n\), the value of an asset is decreased by the fraction \(A_{k}=\frac{n-k+1}{T_{n}}\) of its initial cost, where \(T_{n}=1+2+3+\cdots+n\). (a) If \(n=8\), find \(A_{1}, A_{2}, A_{3}, \ldots, A_{8}\). (b) Show that the sequence in (a) is arithmetic, and find \(S_{8}\). (c) If the initial value of an asset is \(\$ 1000\), how much has been depreciated after 4 years?

Step-by-Step Solution

Answer

After 4 years, the asset depreciates by about $722.22.

Step 1: Calculate Total Sum of Years

The total sum of years, denoted as \( T_n \), is the sum of the first \( n \) natural numbers. The formula for this sum is \( T_n = \frac{n(n+1)}{2} \). For \( n = 8 \), calculate \( T_8 = \frac{8(8+1)}{2} = 36 \).

Step 2: Calculate Depreciation Fractions

Using \( A_k = \frac{n-k+1}{T_n} \), where \( n = 8 \) and \( T_8 = 36 \), compute each fraction: - \( A_1 = \frac{8-1+1}{36} = \frac{8}{36} = \frac{2}{9} \) - \( A_2 = \frac{8-2+1}{36} = \frac{7}{36} \) - \( A_3 = \frac{8-3+1}{36} = \frac{6}{36} = \frac{1}{6} \) - \( A_4 = \frac{8-4+1}{36} = \frac{5}{36} \) - \( A_5 = \frac{8-5+1}{36} = \frac{4}{36} = \frac{1}{9} \) - \( A_6 = \frac{8-6+1}{36} = \frac{3}{36} = \frac{1}{12} \) - \( A_7 = \frac{8-7+1}{36} = \frac{2}{36} = \frac{1}{18} \) - \( A_8 = \frac{8-8+1}{36} = \frac{1}{36} \).

Step 3: Verify Arithmetic Sequence

To determine if the sequence \( A_1, A_2, A_3, \ldots, A_8 \) is arithmetic, find the common difference. The sequence is \( \frac{2}{9}, \frac{7}{36}, \frac{1}{6}, \frac{5}{36}, \frac{1}{9}, \frac{1}{12}, \frac{1}{18}, \frac{1}{36} \). Compute the differences between consecutive terms:- Difference between \( A_2 \) and \( A_1 \) is \( \frac{7}{36} - \frac{2}{9} = -\frac{1}{36} \)- Subsequent terms have the common difference of \( -\frac{1}{36} \), confirming the sequence is arithmetic.

Step 4: Calculate Sum of Arithmetic Sequence

For an arithmetic sequence with common difference \( d = -\frac{1}{36} \) and first term \( a_1 = \frac{2}{9} \), the sum \( S_n \) is given by: \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \). For \( n = 8 \), we have:\[ S_8 = \frac{8}{2} \left( \frac{4}{9} + 7 \times -\frac{1}{36} \right) = 4 \left( \frac{4}{9} - \frac{7}{36} \right) = 4 \times \frac{5}{12} = \frac{20}{12} = \frac{5}{3} \].

Step 5: Calculate Depreciated Value after 4 Years

With initial value \( \$1000 \), calculate depreciation after 4 years using computed fractions:- Sum of first 4 fractions \( A_1 + A_2 + A_3 + A_4 = \frac{2}{9} + \frac{7}{36} + \frac{1}{6} + \frac{5}{36} \)- Convert to common denominator: \( \frac{8}{36} + \frac{7}{36} + \frac{6}{36} + \frac{5}{36} = \frac{26}{36} \)- Depreciation value: \[ \frac{26}{36} \times 1000 = \frac{650}{9} \times 9 \approx 722.22 \text{ (dollars)} \].

Key Concepts

Sum-of-Year's-Digits Method

The Sum-of-Year's-Digits (SYD) Method is a unique approach to calculating depreciation, which represents the loss of value of an asset over time. Unlike straight-line depreciation, where an asset loses the same amount of value each year, SYD uses a decreasing fraction to determine depreciation for each year. This fraction is based on the sum of the years’ digits for the asset's useful life.

In the SYD method, you first calculate the sum of the digits of the years considered. For example, if an asset's useful life is 8 years (as in our original exercise), you add the numbers from 1 to 8, yielding a total of 36. This total is used as the denominator in the fraction for each year's depreciation.

• First year: Largest fraction of depreciation.
• Last year: Smallest fraction of depreciation.

This method is beneficial for those who anticipate that an asset will lose its value more quickly in the early years of use. For businesses, it provides a more realistic figure of depreciation, aligning with how an asset's utility typically diminishes over time.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This concept is important when analyzing the pattern produced by the sum-of-year's-digits method for depreciation.

In the original problem, the depreciation fractions create an arithmetic sequence, because each consecutive fraction decreases by the same amount. For example, in the calculated fractions, the difference between the first two terms, and indeed every pair of consecutive terms thereafter, is \(-\frac{1}{36}\).

• Common Difference: All terms change by the same quantity.
• Sequential Pattern: Each term follows a logical order.

Recognizing this sequence helps in predicting and summing up the depreciation for multiple years, making the SYD method consistent and easy to apply for depreciation calculations.

Asset Valuation

Asset valuation involves determining the current worth of an asset. Depreciation plays an essential role in this process as it reflects the decrease in the asset's value over time. Using the SYD method, the starting value diminishes annually based on calculable fractions.

Initial valuation of an asset is derived from its purchase cost. As each year progresses, that value is reduced by the calculated depreciation amount. For an accurate reflection of business holdings, it's crucial to know both the initial and depreciated values.

• Current Worth: Updated yearly to reflect actual asset worth.
• Investment Decisions: Helps businesses make informed financial choices based on asset depreciation.

Understanding how depreciation affects asset valuation ensures that businesses maintain accurate records and can plan better for future expenses and asset replacements.

Depreciation Calculation

Depreciation calculation refers to the methods used to account for the reduction in value of an asset over time. In the context of the SYD method, each year’s depreciation is calculated using a distinct fractional part of the asset's initial cost.

For example, using the fractions obtained (such as \(\frac{2}{9}\) for the first year) and multiplying these by the asset's original cost (say \$1000 in the exercise), we can determine how much the asset has depreciated at each stage. The total depreciated value after several years can simply be added up from these individual yearly amounts.

• Yearly Calculation: Adjusts the asset's book value annually.
• Accumulated Depreciation: Total decrease over multiple years.

Such precise calculation is invaluable for keeping track of asset value, ensuring accurate balance sheets, and making astute financial decisions regarding asset maintenance and replacement.