To find the 7th partial sum $S_7$ of a geometric sequence, we use the formula for the sum of the first $n$ terms of a geometric sequence:

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

where:

• $a$ is the first term,
• $r$ is the common ratio, and
• $n$ is the number of terms to sum.

Given:

• $a = 5$,
• $r = 3$, and
• $n = 7$,

we can substitute these values into the formula:

$S_7 = \frac{5(1 - 3^7)}{1 - 3}$

Now, calculate $3^7$:

$3^7 = 2187$

Substitute this into the equation:

$S_7 = \frac{5(1 - 2187)}{-2}$ $S_7 = \frac{5(-2186)}{-2}$ $S_7 = \frac{-10930}{-2}$ $S_7 = 5465$

So, the 7th partial sum is $S_7 = 5465$.

Would you like more details or have any questions? Here are some related questions:

1. How does the formula for the sum of a geometric sequence change if $|r| < 1$?
2. What happens to the sum as the number of terms $n$ increases for a geometric sequence with $|r| > 1$?
3. How do you find the sum of an infinite geometric series?
4. How would you find the nth term of a geometric sequence?
5. What is the relationship between the common ratio $r$ and the growth rate of the sequence?

Tip: If the common ratio $r$ is greater than 1, the sequence grows rapidly, which leads to very large sums as $n$ increases.