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This problem is asking to find the sum of a geometric sequence. The given series is:

\[
7 + 7^2 + 7^3 + \cdots + 7^n
\]

This is a geometric series where:
- The first term \( a = 7 \),
- The common ratio \( r = 7 \),
- The number of terms is \( n \).

The formula for the sum of the first \( n \) terms of a geometric series is:

\[
S_n = \frac{a(r^n - 1)}{r - 1}
\]

Substituting the values:
- \( a = 7 \),
- \( r = 7 \),

we get:

\[
S_n = \frac{7(7^n - 1)}{7 - 1} = \frac{7(7^n - 1)}{6}
\]

Thus, the closed form of the sum is:

\[
S_n = \frac{7(7^n - 1)}{6}
\]

Let me know if you'd like further clarification, and feel free to ask additional questions!

### 5 Related Questions:
1. What happens if the first term or the common ratio is a negative number?
2. How does the sum of an infinite geometric series differ from this sum?
3. What are the conditions under which the sum of a geometric series converges?
4. Can you apply this formula to a geometric series with a fractional ratio?
5. How can we derive the formula for the sum of a geometric series?

### Tip:
Always verify if the series you are dealing with is converging or diverging, as it determines whether a finite sum or an infinite sum formula is applicable.

Use the formula for the sum of a geometric sequence to write the following sum in closed form: 7 + 7^2 + 7^3 + ⋯ + 7^n, where n is any integer with n ≥ 1.

Solve the sum of the geometric sequence 7 + 7^2 + 7^3 + ⋯ + 7^n using the geometric series formula. This step-by-step guide covers essential algebra concepts and formulas to derive the closed form of the sum. Suitable for Grades 10-12.

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