7 The sum to $$ n $$ terms of a geometric series is 728 . If the first term is 2 and the last term is 486 , find the common ratio and the number of terms.

Question

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To solve for the common ratio $$ r $$ and the number of terms $$ n $$ in the geometric series, we'll use the given information:

1. **First term ($$ a $$)**: 2
2. **Last term ($$ l $$)**: 486
3. **Sum to $$ n $$ terms ($$ S_n $$)**: 728

### Step 1: Relate the Last Term to the Common Ratio and Number of Terms

The formula for the $$ n $$-th term ($$ l $$) of a geometric series is:
$$
l = a \cdot r^{n-1}
$$
Substituting the known values:
$$
486 = 2 \cdot r^{n-1} \implies r^{n-1} = \frac{486}{2} = 243
$$
Notice that $$ 243 = 3^5 $$, which suggests $$ r = 3 $$ and $$ n-1 = 5 $$, so $$ n = 6 $$.

### Step 2: Verify with the Sum Formula

The sum of the first $$ n $$ terms of a geometric series is given by:
$$
S_n = a \cdot \frac{r^n - 1}{r - 1}
$$
Plugging in $$ a = 2 $$, $$ r = 3 $$, and $$ n = 6 $$:
$$
S_6 = 2 \cdot \frac{3^6 - 1}{3 - 1} = 2 \cdot \frac{729 - 1}{2} = 2 \cdot \frac{728}{2} = 728
$$
This confirms that our values satisfy the given sum.

### Conclusion

- **Common Ratio ($$ r $$)**: 3
- **Number of Terms ($$ n $$)**: 6

**Answer:**  
The common ratio is 3 and the series contains 6 terms.
The common ratio is 3 and there are 6 terms in the series.

7 The sum to terms of a geometric series is 728 . If the first term is 2 and the last term is 486 , find the common ratio and the number of terms.

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Solución

Step 1: Relate the Last Term to the Common Ratio and Number of Terms

Step 2: Verify with the Sum Formula

Conclusion

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