How to Find the Sum of a Geometric Sequence on a Calculator

There are several ways to find the sum of a geometric sequence on a calculator, depending on the type of calculator you have and whether you want an exact sum or an approximate one. Here's how to do it using common calculator functions:

Using the Summation Feature (if available)

Some scientific and graphing calculators have a built-in summation feature (often denoted by the sigma symbol, ∑). This is the most straightforward method.

1. Identify the Formula: Recall the formula for the sum of a geometric series: S_n = a(1 - rⁿ) / (1 - r), where:

   • S_n is the sum of the first n terms
   • a is the first term
   • r is the common ratio
   • n is the number of terms you want to sum

2. Enter the Formula into the Summation Function:

   • Access the summation function on your calculator. This may involve pressing a "math" button or looking in a menu.
   • You'll typically need to define a variable (e.g., x or i) that represents the term number, the starting term number (usually 1), and the ending term number (n).
   • Enter the geometric sequence formula within the summation, replacing the variable with the term number. For example, if you want to sum the sequence 2, 6, 18, 54, ... for the first 5 terms, where a=2 and r=3, you would enter the expression 2 * 3^(x-1) or similar, depending on your calculator's syntax.
   • Specify the starting term number (x=1) and the ending term number (x=5).

3. Evaluate: Press "enter" or "calculate" to get the sum.

Example: Sum the first 5 terms of the geometric sequence 2, 6, 18, 54,... where a=2, r=3 and n=5. You would input the following into the summation function: ∑ (2 * 3^(x-1), x, 1, 5). The calculator will return 242.

Manual Calculation using the Geometric Series Formula

If your calculator lacks a summation function, you can still use the formula.

1. Identify a, r, and n: Determine the first term (a), common ratio (r), and the number of terms (n) you want to sum.

2. Plug the values into the formula: Substitute the values of a, r, and n into the formula S_n = a(1 - rⁿ) / (1 - r).

3. Calculate: Use your calculator to evaluate the expression. Be mindful of the order of operations (PEMDAS/BODMAS).

Example: Using the same sequence (2, 6, 18, 54,... with a=2, r=3, and n=5):

S₅ = 2(1 - 3⁵) / (1 - 3) = 2(1 - 243) / (-2) = 2(-242) / (-2) = -484 / -2 = 242

Iterative Addition (for a small number of terms)

For a very small number of terms, you can simply calculate each term and add them up. This is generally not practical for more than 3-4 terms.

1. Calculate each term individually: a, ar, ar², ar³ ...
2. Add all the terms.

Example: Sum the first three terms of the sequence 2, 6, 18,... 2 + 6 + 18 = 26

Dealing with Infinite Geometric Series

If |r| < 1, you can calculate the sum of an infinite geometric series using the formula S = a / (1 - r). However, this is only valid when the absolute value of the common ratio r is less than 1. Your calculator will simply give you the result after inputting the value.

Example: Find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 +... Here a = 1, and r = 1/2.
S = 1 / (1 - 1/2) = 1 / (1/2) = 2