If the sum of the first n terms of a geometric series is given by S_n = a(1 - r^

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Question: If the sum of the first n terms of a geometric series is given by S_n = a(1 - r^n)/(1 - r), what is the sum when r = 1?

Options:

Correct Answer: na

Solution:

When r = 1, S_n = a(1 - 1^n)/(1 - 1) is indeterminate, but the sum of n terms is na.

If the sum of the first n terms of a geometric series is given by S_n = a(1 - r^n)/(1 - r), what is the sum when r = 1?

If the sum of the first n terms of a geometric series is given by S_n = a(1 - r^n)/(1 - r), what is the sum when r = 1?

Step 1: Understand the formula for the sum of the first n terms of a geometric series, which is S_n = a(1 - r^n)/(1 - r).
Step 2: Identify what happens when r = 1. Substitute r = 1 into the formula: S_n = a(1 - 1^n)/(1 - 1).
Step 3: Simplify the expression. Notice that 1^n is still 1, so the formula becomes S_n = a(1 - 1)/(1 - 1), which is 0/0. This is an indeterminate form.
Step 4: Recognize that when r = 1, each term in the series is just 'a'. Therefore, the first n terms are a + a + a + ... (n times).
Step 5: Calculate the sum of n terms when r = 1. Since there are n terms, the sum is na.

Geometric Series – A geometric series is a series of terms where each term after the first is found by multiplying the previous term by a constant called the common ratio (r).
Sum of Series – The formula for the sum of the first n terms of a geometric series is S_n = a(1 - r^n)/(1 - r), which is valid for r ≠ 1.
Indeterminate Forms – When substituting r = 1 into the sum formula, the expression becomes indeterminate (0/0), requiring a different approach to find the sum.