Example 7 What is the smallest number of terms of the geometrical progression, $$ 8+24+72+\ldots $$, that will give a total greater than 6000000 ?

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To determine the smallest number of terms in the geometric progression $$ 8 + 24 + 72 + \ldots $$ that exceed a total of 6,000,000, follow these steps:

### 1. Identify the Geometric Progression (GP) Parameters:
- **First term ($$ a $$):** 8
- **Common ratio ($$ r $$):** 
  $$
  r = \frac{24}{8} = 3 \quad \text{(since each term is multiplied by 3)}
  $$

### 2. Formula for the Sum of $$ n $$ Terms:
The sum of the first $$ n $$ terms of a GP is given by:
$$
S_n = a \frac{r^n - 1}{r - 1}
$$
Substituting the known values:
$$
S_n = 8 \frac{3^n - 1}{3 - 1} = 4 (3^n - 1)
$$

### 3. Set Up the Inequality:
We want $$ S_n $$ to be greater than 6,000,000:
$$
4 (3^n - 1) > 6,000,000
$$
Divide both sides by 4:
$$
3^n - 1 > 1,500,000
$$
Add 1 to both sides:
$$
3^n > 1,500,001
$$

### 4. Solve for $$ n $$:
To find the smallest integer $$ n $$ that satisfies the inequality, take the logarithm of both sides:
$$
n \log(3) > \log(1,500,001)
$$
$$
n > \frac{\log(1,500,001)}{\log(3)} \approx \frac{6.1761}{0.4771} \approx 12.94
$$
Since $$ n $$ must be an integer, round up to the next whole number:
$$
n = 13
$$

### 5. Verification:
- **For $$ n = 13 $$:**
  $$
  3^{13} = 1,594,323
  $$
  $$
  S_{13} = 4 (1,594,323 - 1) = 4 \times 1,594,322 = 6,377,288 > 6,000,000
  $$
  
- **For $$ n = 12 $$:**
  $$
  3^{12} = 531,441
  $$
  $$
  S_{12} = 4 (531,441 - 1) = 4 \times 531,440 = 2,125,760 < 6,000,000
  $$

Thus, **13 terms** are needed to exceed a total of 6,000,000.

### **Answer:**  
13
13 terms are needed to exceed a total of 6,000,000.

Example 7 What is the smallest number of terms of the geometrical progression, \( 8+24+72+\ldots \), that will give a total greater than 6000000 ?

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