The problem in the image asks to evaluate the sum:

$\sum_{n=1}^{20} 3^{n-2}.$

Let’s solve this step by step:

Step 1: General form of the sum

The term $3^{n-2}$ is a geometric sequence where the first term ($a$) and the common ratio ($r$) need to be identified:

• For $n = 1$: $3^{n-2} = 3^{1-2} = 3^{-1} = \frac{1}{3}$, so $a = \frac{1}{3}$.
• The common ratio $r = 3$ (since each term is multiplied by 3).

Step 2: Formula for the sum of a geometric series

The formula for the sum of the first $N$ terms of a geometric series is:

$S_N = a \frac{1 - r^N}{1 - r}, \quad r \neq 1.$

Here:

• $a = \frac{1}{3}$,
• $r = 3$,
• $N = 20$.

Step 3: Substitute the values into the formula

$S_{20} = \frac{1}{3} \cdot \frac{1 - 3^{20}}{1 - 3}.$

Simplify the denominator: $1 - 3 = -2.$

Thus: $S_{20} = \frac{1}{3} \cdot \frac{1 - 3^{20}}{-2}.$

Simplify further: $S_{20} = \frac{1 - 3^{20}}{-6}.$

Step 4: Rearrange the expression

Distribute the negative sign: $S_{20} = \frac{3^{20} - 1}{6}.$

Final Answer:

$\boxed{S_{20} = \frac{3^{20} - 1}{6}}$

Would you like a detailed explanation of any part of this solution or a numerical approximation?

Related Questions:

1. How do you derive the formula for the sum of a geometric series?
2. What is the value of $3^{20}$ approximately?
3. How would the solution change if the sum started from $n = 0$ instead of $n = 1$?
4. What are some real-life applications of geometric series?
5. Can you generalize the sum for any $n$ terms of $a^n$ in terms of $a$ and $r$?

Tip:

Always confirm the first term and common ratio carefully when working with geometric series!