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.