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), which of the following is true?

Options:

Correct Answer: S_n can be negative.

Solution:

The sum S_n can be negative if the common ratio r is negative and the first term a is also negative.

If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n)/(1 - r), which of the following is true?

If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n)/(1 - r), which of the following is true?

Step 1: Understand what a geometric progression (GP) is. A GP is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).
Step 2: Know the formula for the sum of the first n terms of a GP, which is S_n = a(1 - r^n)/(1 - r). Here, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Step 3: Recognize that the value of S_n can change based on the values of 'a' and 'r'.
Step 4: If 'r' is negative, it means the terms of the GP will alternate in sign (positive, negative, positive, etc.).
Step 5: If 'a' (the first term) is also negative, the first term will start the sequence as negative.
Step 6: When you sum these alternating terms, the total sum S_n can end up being negative, depending on how many terms you are adding and the values of 'a' and 'r'.
Step 7: Therefore, it is true that S_n can be negative if both 'a' is negative and 'r' is negative.

Geometric Progression – A sequence of numbers 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 a Geometric Series – The formula for the sum of the first n terms of a geometric series, which can vary based on the values of the first term and the common ratio.
Behavior of S_n – Understanding how the sum S_n can be influenced by the signs of the first term and the common ratio.