Master Geometric Progression applications for CBSE Class 11 Applied Mathematics. Learn compound interest, population growth, and series sum problems.
Geometric progressions are fundamental in modeling financial growth, specifically compound interest and depreciation. When a principal amount P grows at a constant rate r per period, the amount after n periods forms a GP. This allows for the calculation of future values and total accumulated wealth over time.
Step 1: Identify P = 1000, r = 0.10, n = 3.
Step 2: Use the formula A = 1000(1 + 0.10)³.
Step 3: Calculate 1000 × 1.331.
Answer: 1331
Population growth often follows a geometric pattern where the population at the end of a year is a fixed multiple of the previous year. If a population starts at P₀ and grows at a rate of r percent, the population after n years is calculated using the nth term of a GP. This model is essential for demographic forecasting.
Step 1: Given P₀ = 5000, r = 5%, n = 2.
Step 2: P₂ = 5000(1 + 0.05)².
Step 3: 5000 × 1.1025 = 5512.5.
Answer: 5513 (approx)
Many physical phenomena, such as the total distance covered by a bouncing ball, involve the sum of an infinite geometric series. If a ball drops from height H and rebounds to a fraction r of its previous height, the total distance is the sum of the series. This requires the formula for the sum of an infinite GP where |r| < 1.
Step 1: H = 10m, r = 0.5.
Step 2: Total distance = H + 2(Hr + Hr² + ...).
Step 3: 10 + 2(5 / (1 - 0.5)) = 10 + 20 = 30.
Answer: 30 meters