Calculate nth term, sum of series, infinite series, and geometric sequence properties with step-by-step solutions. Perfect for mathematics, algebra, and series analysis.
Comprehensive geometric sequence calculations with detailed explanations
Find any term in a geometric sequence using the formula aₙ = a₁ × r^(n-1)
Calculate the sum of finite and infinite geometric series with convergence analysis
Find common ratio, first term, and analyze convergence properties
Detailed explanations and mathematical methods for every calculation
Understanding geometric sequences and series calculations
Input the first term, common ratio, term number, or other required values for your geometric sequence calculation.
Choose from nth term, sum of series, infinite series, finding common ratio, or finding first term calculations.
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aₙ = a₁ × r^(n-1)
Sₙ = a₁(1 - r^n)/(1 - r)
S∞ = a₁/(1 - r) when |r| < 1
Common geometric sequence patterns and their applications
5th term = 2 × 3^4 = 162
Sum of first 5 terms = 1(1-2^5)/(1-2) = 31
Infinite sum = 1/(1-1/2) = 2
Infinite sum = 100/(1-0.5) = 200
Common questions about geometric sequences and series
A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. The general form is a₁, a₁r, a₁r², a₁r³, ...
Use the formula aₙ = a₁ × r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number you want to find.
For a finite geometric series with n terms, the sum is Sₙ = a₁(1 - r^n)/(1 - r) when r ≠ 1. For an infinite geometric series, the sum is S∞ = a₁/(1 - r) when |r| < 1.
An infinite geometric series converges when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges.
In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.
Divide any term by the previous term: r = aₙ/aₙ₋₁. Alternatively, if you know the first term and any nth term, you can use r = (aₙ/a₁)^(1/(n-1)).
Yes, the common ratio can be negative. This creates an alternating geometric sequence where terms alternate between positive and negative values.
Geometric sequences are used in compound interest calculations, population growth models, radioactive decay, computer algorithms, and many other exponential growth or decay scenarios.
Rearrange the nth term formula: a₁ = aₙ / r^(n-1). This gives you the first term when you know the common ratio, term number, and the value of that term.
Geometric sequences are discrete versions of exponential functions. The nth term formula aₙ = a₁ × r^(n-1) is similar to the exponential function f(x) = a × r^x.
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