Home Class 11 Applied Maths Chapter 6: Sequences and Series Exercise 6.4: Practical and Business Applications
Chapter 6 • Unit II

Exercise 6.4: Practical and Business Applications

Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).

6.4 Practical and Business Applications of Sequences and Series

Applied Mathematics bridges the abstract algebra of progressions with the operational reality of enterprise economics, accounting standards, and demographic forecasting. This chapter consolidates the practical applications of Arithmetic Progressions (linear models) and Geometric Progressions (exponential models) in asset depreciation, compensation ladders, compound investment plans, and macroeconomic multipliers.

Comprehensive Financial & Business Modeling Framework

1. Capital Asset Depreciation: Linear vs Compounded Models

Attribute Straight Line Method (SLM) — AP Written Down Value (WDV) — GP
Mathematical Nature Arithmetic Progression (Constant annual deduction) Geometric Progression (Constant annual percentage decay)
Book Value Formula (Year \(n\)) \[ V_n = V_0 - n \cdot D \] \[ V_n = V_0 (1 - r)^n \]
Depreciation in Year \(n\) \[ D = \frac{V_0 - S}{L} \quad (\text{Constant}) \] \[ D_n = r \cdot V_{n-1} = r V_0 (1 - r)^{n-1} \]
Scrap Value Reachability Reaches exact scrap value \(S\) at end of useful life \(L\) Asymptotically approaches zero; never theoretically reaches absolute zero

2. Structured Corporate Compensation Ladders

An employment salary grade with an initial pay \(a\) and fixed annual increment \(d\) represents an AP: \[ \text{Salary in Year } n = a + (n - 1)d \] \[ \text{Total Earnings over } n \text{ Years} = \frac{n}{2} [2a + (n - 1)d] \]

3. Population and Demographic Projections

Demographic counts growing at an annual compounding growth rate \(r\) follow an exponential GP: \[ P_n = P_0 (1 + r)^n \] If growth rate varies year-over-year: \(P_n = P_0 (1 + r_1)(1 + r_2)\dots(1 + r_n)\).

Comprehensive Financial Worked Example: Corporate Asset Replacement Strategy

Scenario: A manufacturing enterprise purchases heavy industrial machinery for ₹5,00,000. The operational lifespan of the machine is estimated to be 5 years.

  1. If the company adopts the Straight Line Method (SLM) with an expected scrap value of ₹50,000, find the annual depreciation charge and the book value at the end of the 3rd year.
  2. If the company adopts the Written Down Value (WDV) Method at a statutory depreciation rate of 20% per annum, find the book value at the end of 3 years and the depreciation charge for the 4th year.

Solution:

1. Under the Straight Line Method (SLM):

  • Cost \(V_0 = ₹5{,}00{,}000\), Scrap value \(S = ₹50{,}000\), Lifespan \(L = 5\) years.
  • Annual constant depreciation: \[ D = \frac{V_0 - S}{L} = \frac{5{,}00{,}000 - 50{,}000}{5} = \frac{4{,}50{,}000}{5} = ₹90{,}000 \text{ per annum.} \]
  • Book value at the end of 3 years (using AP): \[ V_3 = V_0 - 3D = 5{,}00{,}000 - 3(90{,}000) = 5{,}00{,}000 - 2{,}70{,}000 = ₹2{,}30{,}000. \]

2. Under the Written Down Value (WDV) Method:

  • Depreciation rate \(r = 20\% = 0.20\).
  • Book value at the end of 3 years (using GP): \[ V_3 = V_0 (1 - r)^3 = 5{,}00{,}000 \times (1 - 0.20)^3 = 5{,}00{,}000 \times (0.80)^3 \] \[ V_3 = 5{,}00{,}000 \times 0.512 = ₹2{,}56{,}000. \]
  • Depreciation charge during the 4th year: \[ D_4 = r \times V_3 = 0.20 \times 2{,}56{,}000 = ₹51{,}200. \]
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