Master the sum of n terms and infinite terms of a Geometric Progression with clear formulas, solved examples, and practice problems for CBSE Class 11.
The sum of the first n terms of a Geometric Progression (GP) with first term 'a' and common ratio 'r' is determined by the relationship between 'r' and 1. When r ≠ 1, the sum is calculated using the formula involving the number of terms n. This formula is essential for financial mathematics, such as calculating the future value of an annuity.
Step 1: Identify a = 2 and r = 6/2 = 3.
Step 2: Since r > 1, use Sₙ = a(rⁿ − 1) / (r − 1).
Step 3: Substitute n = 5: S₅ = 2(3⁵ − 1) / (3 − 1).
Step 4: Calculate 3⁵ = 243, so S₅ = 2(242) / 2.
Answer: S₅ = 242.
When the common ratio 'r' satisfies the condition |r| < 1, the sum of an infinite GP converges to a finite value. As n approaches infinity, the term rⁿ approaches 0, simplifying the sum formula significantly. This concept is vital in understanding repeating decimals and convergent series.
Step 1: Identify a = 1 and r = 1/2.
Step 2: Check condition |1/2| < 1, which is true.
Step 3: Apply S∞ = a / (1 − r).
Step 4: S∞ = 1 / (1 − 0.5) = 1 / 0.5.
Answer: S∞ = 2.
The sum of a GP is sensitive to the value of the common ratio. If r = 1, the sequence is constant (a, a, a, ...), and the sum of n terms is simply n × a. If |r| ≥ 1, the sum of infinite terms does not exist as it diverges to infinity.
Step 1: Identify a = 5 and r = 1.
Step 2: Use the formula Sₙ = n × a.
Step 3: Substitute n = 10 and a = 5.
Step 4: S₁₀ = 10 × 5.
Answer: S₁₀ = 50.