What is the relationship between the first term and the common ratio if the sum

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Question: What is the relationship between the first term and the common ratio if the sum of an infinite GP converges?

Options:

Correct Answer: The common ratio must be less than one in absolute value.

Solution:

For the sum of an infinite GP to converge, the common ratio must be less than one in absolute value.

What is the relationship between the first term and the common ratio if the sum of an infinite GP converges?

What is the relationship between the first term and the common ratio if the sum of an infinite GP converges?

Step 1: Understand what a GP (Geometric Progression) is. It is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio.
Step 2: Identify the first term of the GP, which we can call 'a'.
Step 3: Recognize the common ratio, which we can call 'r'.
Step 4: Learn that the sum of an infinite GP can be calculated using the formula S = a / (1 - r), but this formula only works if certain conditions are met.
Step 5: For the sum to converge (meaning it approaches a specific value), the absolute value of the common ratio must be less than 1. This means |r| < 1.
Step 6: If |r| is less than 1, the terms of the GP get smaller and smaller, allowing the sum to settle at a finite value.
Step 7: If |r| is 1 or greater, the terms do not get smaller, and the sum does not converge.

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