Geometric Sequence ಅನ್ನು ಹೇಗೆ ಲೆಕ್ಕ ಹಾಕುವುದು
Geometric Sequence ಎಂದರೇನು?
A geometric sequence has each term as a constant multiple (common ratio r) of the previous term. Geometric sequences model exponential growth, compound interest, and population dynamics.
ಸೂತ್ರ
nth term: aₙ = a₁ × rⁿ⁻¹; Sum of n terms: Sₙ = a₁(1−rⁿ)/(1−r)
- a₁
- first term
- r
- common ratio — ratio of each term to the previous
- n
- number of terms
- aₙ
- nth term
- Sₙ
- sum of first n terms
ಹಂತ-ಹಂತದ ಮಾರ್ಗದರ್ಶಿ
- 1aₙ = a₁ × rⁿ⁻¹
- 2Sum of n terms: Sₙ = a₁(1−rⁿ)/(1−r)
- 3Sum to infinity (|r|<1): S∞ = a₁/(1−r)
- 4Ratio r = aₙ/aₙ₋₁
Worked Examples
ಇನ್ಪುಟ್
a₁ = 2, r = 3, 5 terms
ಫಲಿತಾಂಶ
2, 6, 18, 54, 162 — Sum = 242
ಇನ್ಪುಟ್
a₁ = 1, r = 0.5, ∞ terms
ಫಲಿತಾಂಶ
Sum to infinity = 1/(1−0.5) = 2
Frequently Asked Questions
What is the difference between arithmetic and geometric sequences?
Arithmetic: constant difference between consecutive terms. Geometric: constant ratio between consecutive terms.
What happens in a geometric sequence if r = 1?
All terms are identical. The sequence is constant: a, a, a, a, ...
What is an infinite geometric series?
If |r| < 1, the infinite sum converges: S∞ = a₁/(1−r). If |r| ≥ 1, the series diverges.
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