Master the nth term of a Geometric Progression with our comprehensive CBSE Class 11 Applied Mathematics guide, featuring solved examples and practice.
A Geometric Progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the preceding term by a fixed non-zero number called the common ratio. If a is the first term and r is the common ratio, the sequence is represented as a, ar, ar², ar³, .... This progression is fundamental in modeling exponential growth and decay in financial mathematics.
Step 1: Identify a = 2 and r = 3.
Step 2: Use formula a₆ = 2·3⁶⁻¹.
Step 3: Calculate 2·3⁵ = 2·243 = 486.
Answer: 486
The nth term formula allows us to find any specific position in the sequence without listing all preceding terms. The common ratio r can be calculated by dividing any term by its immediate predecessor, r = aₙ/aₙ₋₁. If r > 1, the terms increase in magnitude; if 0 < r < 1, the terms decrease toward zero.
Step 1: Given sequence 81, 27, 9, 3.
Step 2: Divide second term by first: 27/81 = 1/3.
Step 3: Verify with third term: 9/27 = 1/3.
Answer: 1/3
When the first term, common ratio, and the last term are known, we can determine the total number of terms in the sequence. By rearranging the nth term formula, we solve for n using logarithmic properties or simple algebraic manipulation. This is essential for calculating the duration of investment growth.
Step 1: a = 3, r = 2, aₙ = 384.
Step 2: 384 = 3·2ⁿ⁻¹ ⟹ 128 = 2ⁿ⁻¹.
Step 3: 2⁷ = 2ⁿ⁻¹ ⟹ n-1 = 7 ⟹ n = 8.
Answer: 8