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Identify the First Term and Common Ratio
First, identify the first term (a) and the common ratio (r) of the geometric sequence. These values are crucial in calculating the nth term and partial sum.
Choose the Correct Formula
Next, choose the correct formula depending on what you want to calculate. Use the formula an = ar^(n-1) to find the nth term, and the formula Sn = a * (1 - r^n) / (1 - r) to find the partial sum.
Plug in the Values
Plug in the values of a, r, and n into the chosen formula. Make sure to follow the order of operations (PEMDAS) to avoid mistakes.
Simplify the Expression
Simplify the expression by performing the necessary calculations. This may involve exponentiation, multiplication, and division.
Check for Convergence
If the common ratio is between -1 and 1 (exclusive), the geometric sequence converges. You can use the formula for the sum of an infinite geometric series: S = a / (1 - r).
Use a Calculator for Convenience
While manual calculations are possible, using a geometric sequence calculator can be convenient for complex calculations or when dealing with large numbers. It can also help you avoid common mistakes such as incorrect exponentiation or division.
Introduction to Geometric Sequences
A geometric sequence is a type of sequence where each term is found by multiplying the previous term by a constant factor, known as the common ratio. In this guide, we will learn how to calculate the nth term and partial sum of a geometric sequence manually.
Understanding the Formula
The formula for the nth term of a geometric sequence is: an = ar^(n-1) where a is the first term and r is the common ratio.
The formula for the partial sum of a geometric sequence is: Sn = a * (1 - r^n) / (1 - r) when r ≠ 1.
Worked Example
Let's say we have a geometric sequence with a first term of 2 and a common ratio of 3. We want to find the 4th term and the sum of the first 4 terms.
To find the 4th term, we plug in the values into the formula: a4 = 2 * 3^(4-1) = 2 * 3^3 = 2 * 27 = 54
To find the sum of the first 4 terms, we plug in the values into the formula: S4 = 2 * (1 - 3^4) / (1 - 3) = 2 * (1 - 81) / (-2) = 2 * (-80) / (-2) = 80