Determine $$ m $$ in each of the geometric series: a) $$ 3+6+12+\ldots $$ (to $$ m $$ terms) $$ =93 $$

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To determine the number of terms $$ m $$ in the geometric series $$ 3 + 6 + 12 + \ldots = 93 $$, follow these steps:

1. **Identify the First Term and Common Ratio:**
   - First term ($$ a $$) = 3
   - Common ratio ($$ r $$) = $$ \frac{6}{3} = 2 $$

2. **Use the Formula for the Sum of a Geometric Series:**
   The sum of the first $$ m $$ terms of a geometric series is given by:
   $$
   S_m = a \frac{r^m - 1}{r - 1}
   $$
   Plugging in the known values:
   $$
   93 = 3 \frac{2^m - 1}{2 - 1}
   $$
   Simplify:
   $$
   93 = 3 (2^m - 1)
   $$

3. **Solve for $$ m $$:**
   $$
   93 = 3 \times 2^m - 3
   $$
   $$
   93 + 3 = 3 \times 2^m
   $$
   $$
   96 = 3 \times 2^m
   $$
   $$
   32 = 2^m
   $$
   Since $$ 2^5 = 32 $$:
   $$
   m = 5
   $$

**Answer:**  
Problem a Answer:

The number of terms $$ m $$ is $$ 5 $$.
The number of terms $$ m $$ is 5.

Determine in each of the geometric series:
a) (to terms)

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Determine in each of the geometric series: a) (to terms)