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If the sum of the first n terms of a geometric progression is given by S_n = a(1

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Question: If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n) / (1 - r), what happens to S_n as n approaches infinity when |r| < 1?

Options:

  1. S_n approaches 0
  2. S_n approaches infinity
  3. S_n approaches a/(1-r)
  4. S_n approaches a

Correct Answer: S_n approaches a/(1-r)

Solution: 

As n approaches infinity and |r| < 1, r^n approaches 0, thus S_n approaches a/(1-r).

If the sum of the first n terms of a geometric progression is given by S_n = a(1

Practice Questions

Q1
If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n) / (1 - r), what happens to S_n as n approaches infinity when |r| < 1?
  1. S_n approaches 0
  2. S_n approaches infinity
  3. S_n approaches a/(1-r)
  4. S_n approaches a

Questions & Step-by-Step Solutions

If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n) / (1 - r), what happens to S_n as n approaches infinity when |r| < 1?
  • Step 1: Understand the formula for the sum of the first n terms of a geometric progression: S_n = a(1 - r^n) / (1 - r).
  • Step 2: Identify the condition given in the question: |r| < 1. This means that the absolute value of r is less than 1.
  • Step 3: Recognize what happens to r^n as n becomes very large (approaches infinity). Since |r| < 1, r^n gets smaller and smaller, approaching 0.
  • Step 4: Substitute the limit of r^n into the formula for S_n. As n approaches infinity, r^n approaches 0, so S_n becomes S_n = a(1 - 0) / (1 - r).
  • Step 5: Simplify the expression: S_n = a / (1 - r). This is the value S_n approaches as n approaches infinity.
  • Geometric Progression – A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Sum of Infinite Series – The sum of the terms of a series as the number of terms approaches infinity, particularly relevant for geometric series with a common ratio less than 1 in absolute value.
  • Limit Behavior – Understanding how a function behaves as its input approaches a certain value, in this case, how S_n behaves as n approaches infinity.
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