The sum to infinity of a GP is 100, find its first term if the common ratio is -$\frac{1}{2}$

1. Analyze the question

We are given the sum to infinity ($S_\infty$) of a Geometric Progression (GP) and the common ratio ($r$). We need to find the first term ($a$).

2. Explain the concept

For a Geometric Progression where the common ratio $|r| < 1$, the sum to infinity is calculated using the formula:

Where:

• $S_\infty$ = Sum to infinity
• $a$ = First term
• $r$ = Common ratio

3. Show the solution

Given:

• $S_\infty = 100$
• $r = -\frac{1}{2}$

Step 1: Substitute the values into the formula

Step 2: Simplify the denominator

Step 3: Solve for $a$

Multiply both sides by 1.5:

4. Identify the answer

The calculated first term is 150, which corresponds to option A.

5. Tip

When dealing with a negative common ratio in the sum to infinity formula, always be careful with the signs in the denominator ($1 - r$). Many students mistakenly add the ratio directly instead of subtracting a negative, which leads to $1 - 0.5 = 0.5$ instead of the correct $1 + 0.5 = 1.5$. Always write out the signs clearly to avoid this common JAMB trap!