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?
Practice Questions
1 question
Q1
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?
na
a
0
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When r = 1, S_n = a(1 - 1^n)/(1 - 1) is indeterminate, but the sum of n terms is na.
Questions & Step-by-step Solutions
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Q
Q: 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?
Solution: When r = 1, S_n = a(1 - 1^n)/(1 - 1) is indeterminate, but the sum of n terms is na.
Steps: 5
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.
