Master Geometric Progressions for CBSE Class 11 Applied Mathematics. Learn formulas, common ratios, and solve step-by-step problems with ease.
A Geometric Progression (GP) is a sequence of non-zero numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio. If the first term is a and the common ratio is r, the sequence is represented as a, ar, ar², ar³, .... This progression is fundamental in modeling exponential growth and decay in finance and population studies.
Step 1: Given sequence 3, 6, 12, 24.
Step 2: Check ratio r = 6/3 = 2 and 12/6 = 2.
Step 3: Since the ratio is constant, it is a GP.
Answer: First term a=3, common ratio r=2.
The n-th term of a GP allows us to calculate any specific value in the sequence without listing all preceding terms. By substituting the first term a, the common ratio r, and the position n into the standard formula, we can determine the magnitude of any term. This is particularly useful in calculating compound interest or depreciating asset values over time.
Step 1: Given a=2, r=3, n=6.
Step 2: Apply formula a₆ = 2·3⁶⁻¹.
Step 3: Calculate 3⁵ = 243.
Step 4: a₆ = 2·243 = 486.
Answer: 486.
The common ratio r is determined by dividing any term by its preceding term, i.e., r = aₙ/aₙ₋₁. If r > 1, the terms increase in magnitude; if 0 < r < 1, the terms decrease; and if r is negative, the terms alternate in sign. Understanding these properties is essential for analyzing the behavior of sequences in applied mathematics.
Step 1: Identify consecutive terms: a₁=100, a₂=-50.
Step 2: Calculate r = -50/100.
Step 3: Simplify the fraction.
Answer: r = -0.5.