Home Class 11 Applied Maths Chapter 6: Sequences and Series Exercise 6.2: Geometric Progression (GP)
Chapter 6 • Unit II

Exercise 6.2: Geometric Progression (GP)

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

6.2 Geometric Progression: Theory, Infinite Series, and Financial Multipliers

In Applied Mathematics, corporate finance, and macroeconomics, a Geometric Progression (GP) represents multiplicative growth or decay where each term is generated by multiplying the preceding term by a fixed common ratio \(r\). GPs provide the quantitative foundation for compound interest, annuity accumulation, declining-balance asset depreciation, viral marketing coefficients, and the Keynesian expenditure multiplier.

Fundamental Formulae & Theorems

  • Definition: A sequence \(a_1, a_2, a_3, \dots\) is a GP if: \[ \frac{a_{k+1}}{a_k} = r \quad (\text{constant common ratio for all } k \ge 1, \, r \neq 0) \]
  • General Term (\(n\)-th Term): \[ a_n = a r^{n - 1} \] where \(a = a_1\) is the non-zero first term and \(r\) is the common ratio.
  • \(n\)-th Term from the End: \[ l_n = \frac{l}{r^{n - 1}} \quad (\text{where } l \text{ is the last term}) \]
  • Sum of First \(n\) Terms (\(S_n\)): \[ S_n = \begin{cases} \displaystyle \frac{a(1 - r^n)}{1 - r} = \frac{a(r^n - 1)}{r - 1}, & \text{if } r \neq 1 \\[10pt] n a, & \text{if } r = 1 \end{cases} \]
  • Sum of an Infinite GP (\(S_\infty\)): If and only if the absolute common ratio is strictly less than unity (\(|r| < 1\)): \[ S_\infty = \frac{a}{1 - r} \] Economic Note: This infinite series formula directly derives the Macroeconomic Keynesian Multiplier: \(k = \frac{1}{1 - \text{MPC}}\) where MPC is the Marginal Propensity to Consume.
  • Geometric Mean (GM): For any two positive real numbers \(a\) and \(b\): \[ \text{GM} = \sqrt{ab} \]
  • Insertion of \(n\) Geometric Means between \(a\) and \(b\): The common ratio is: \[ r = \left( \frac{b}{a} \right)^{\frac{1}{n + 1}} \] The \(k\)-th inserted mean is: \[ G_k = a \cdot r^k = a \left( \frac{b}{a} \right)^{\frac{k}{n + 1}} \] The product of \(n\) inserted geometric means equals the \(n\)-th power of their single GM: \[ \prod_{k=1}^n G_k = (\sqrt{ab})^n = (ab)^{\frac{n}{2}} \]
  • Symmetric Choice of Variables:
    • 3 terms in GP: \(\frac{a}{r}, \, a, \, ar\) (product \(= a^3\))
    • 4 terms in GP: \(\frac{a}{r^3}, \, \frac{a}{r}, \, ar, \, ar^3\) (common ratio \(r^2\), product \(= a^4\))
    • 5 terms in GP: \(\frac{a}{r^2}, \, \frac{a}{r}, \, a, \, ar, \, ar^2\) (product \(= a^5\))

Applied Economic Case: The Multiplier Effect

Example: National Income Generation via Capital Injections

Scenario: The government initiates an infrastructure project with an initial direct investment of ₹10,000 Crores. In this economy, consumers spend 80% of any additional income they receive (Marginal Propensity to Consume, \(\text{MPC} = 0.8\)). Assuming consecutive rounds of expenditure, calculate:

  1. The incremental income generated in the 4th round.
  2. The total cumulative increase in national income generated by this investment.

Solution:

The sequence of expenditures across rounds forms an infinite GP:

\[ a = 10{,}000 \text{ Crores}, \quad r = \text{MPC} = 0.8 = \frac{4}{5} \]
  1. Incremental income in Round 4 (\(a_4\)): \[ a_4 = a \cdot r^3 = 10{,}000 \times (0.8)^3 = 10{,}000 \times 0.512 = 5{,}120 \text{ Crores.} \]
  2. Total cumulative national income (\(S_\infty\)): Since \(|r| = 0.8 < 1\), the infinite series converges: \[ S_\infty = \frac{a}{1 - r} = \frac{10{,}000}{1 - 0.8} = \frac{10{,}000}{0.2} = 50{,}000 \text{ Crores.} \] Economic Takeaway: The initial ₹10,000 Crore investment creates a total economic expansion of ₹50,000 Crores (a 5-fold multiplier effect).
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