요금제

등비수열 계산기 (일반항과 합)

이 계산기는 각 항이 앞의 항에 같은 수를 곱한 값인 수열의 제n항과 첫째항부터 제n항까지의 합을 구합니다. 공비가 −1과 1 사이이면 무한등비급수의 합도 구합니다.

결과

값을 입력하고 계산을 누르면 결과가 표시됩니다.

공식

an=a1rn−1a_n = a_1 r^{n-1}

Multiply the first term by the ratio n − 1 times.

Sn=a1(1−rn)1−r,r≠1S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad r \neq 1

The sum of the first n terms.

S∞=a11−r,∣r∣<1S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1

The sum to infinity exists only when the terms shrink towards 0.

이 계산기 사용법

  1. Enter the first term and the common ratio (divide any term by the one before it).
  2. Enter n, the term you want (and how many terms to add).
  3. Press Calculate to see the nth term, the sum, the first terms and the infinite sum if it exists.

풀이 예제

2, 6, 18, … : the 5th term and the sum of 5 terms

a₅ = 2 × 3⁴ = 162.

S₅ = 2(1 − 3⁵)/(1 − 3) = 2 × (−242)/(−2) = 242.

r = 3 > 1, so there is no sum to infinity.

8, 4, 2, … : halving each time

a₄ = 8 × (1/2)³ = 1.

S₄ = 8 + 4 + 2 + 1 = 15.

Sum to infinity: 8/(1 − 1/2) = 16.

Alternating signs: 5, −10, 20, …

a₆ = 5 × (−2)⁵ = −160.

S₆ = 5(1 − 64)/(1 + 2) = −105.

자주 하는 실수

  • Using rⁿ instead of rⁿ⁻¹ for the nth term.
  • Finding r by subtracting terms (that is the arithmetic difference); divide instead.
  • Using the sum to infinity when |r| ≥ 1: the sum grows without limit.

자주 묻는 질문

How do you find the common ratio?

Divide a term by the one before it: 6 ÷ 2 = 3. The ratio is the same for every pair.

When does a geometric series have a sum to infinity?

Only when −1 < r < 1, so the terms get smaller and smaller. Then S∞ = a₁/(1 − r).

What is the difference between arithmetic and geometric sequences?

An arithmetic sequence adds the same number each time (3, 7, 11); a geometric sequence multiplies by the same number (2, 6, 18).