Master application problems of AP and GP for CBSE Class 11 Applied Mathematics. Learn to solve real-world financial and growth models with step-by-step guides.
Arithmetic Progressions (AP) are used to model scenarios with constant periodic changes, such as simple interest or fixed annual salary increments. If a value changes by a fixed amount d each period, the value at time n is given by the formula for the nth term. This linear growth is fundamental for calculating total savings over time.
Step 1: Identify a = 20000, d = 1500, n = 5.
Step 2: Apply a₅ = 20000 + (5 − 1)1500.
Step 3: Calculate 20000 + 6000 = 26000.
Answer: The salary in the 5th year is ₹26000.
Geometric Progressions (GP) model exponential growth or decay, such as compound interest, population growth, or depreciation of assets. When a quantity increases by a fixed percentage rate r, the value follows a GP where the common ratio is (1 + r/100). These models are essential for long-term financial planning.
Step 1: Given a = 1000, r = 1.1 (10% growth), n = 3.
Step 2: Apply a₃ = 1000·(1.1)³⁻¹.
Step 3: Calculate 1000·1.21 = 1210.
Answer: The population after 2 years is 1210.
Summation formulas allow us to calculate the total accumulation of funds over a specific duration. For AP, we use the sum of n terms to find total deposits, while for GP, we use the sum formula to find the future value of an annuity or total investment growth. These tools are vital for evaluating financial instruments.
Step 1: Given a = 1000, r = 2, n = 4.
Step 2: Apply S₄ = 1000(2⁴ − 1)/(2 − 1).
Step 3: Calculate 1000(15)/1 = 15000.
Answer: The total sum is 15000.