الخطط

حاسبة المتتالية الهندسية (الحد النوني والمجموع)

توجد هذه الآلة الحاسبة الحد النوني ومجموع أول n حدًا لمتتالية كل حد فيها يساوي الحد السابق مضروبًا في العدد نفسه. عندما تكون النسبة بين −1 و1 تعطي أيضًا المجموع إلى ما لا نهاية.

النتيجة

أدخل القيم واضغط احسب لرؤية النتيجة.

الصيغة

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.

كيفية استخدام هذه الحاسبة

  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.

أمثلة محلولة

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.

أخطاء شائعة

  • 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.

الأسئلة الشائعة

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).