The provided image contains a mathematical expression. Let me transcribe it:

It asks for "a soma de" (the sum of): $\sum_{i=1}^{2022} \binom{2023}{i}$

Solution:

The given expression is the sum of binomial coefficients from $i = 1$ to $i = 2022$, where $\binom{2023}{i}$ represents the binomial coefficient, defined as:

$\binom{2023}{i} = \frac{2023!}{i! (2023-i)!}$

Property of Binomial Coefficients:

There is a well-known property of binomial coefficients:

$\sum_{i=0}^{n} \binom{n}{i} = 2^n$

For this specific case, $n = 2023$, so:

$\sum_{i=0}^{2023} \binom{2023}{i} = 2^{2023}$

However, the question only asks for the sum from $i = 1$ to $i = 2022$, excluding the terms for $i = 0$ and $i = 2023$. The binomial coefficients for $i = 0$ and $i = 2023$ are both equal to 1 ($\binom{2023}{0} = \binom{2023}{2023} = 1$).

Thus:

$\sum_{i=1}^{2022} \binom{2023}{i} = 2^{2023} - \binom{2023}{0} - \binom{2023}{2023}$

Substituting $\binom{2023}{0} = 1$ and $\binom{2023}{2023} = 1$:

$\sum_{i=1}^{2022} \binom{2023}{i} = 2^{2023} - 1 - 1 = 2^{2023} - 2$

Final Answer:

The sum is:

$\boxed{2^{2023} - 2}$

Let me know if you'd like a detailed explanation of any part of the solution! 😊

Follow-up Questions:

1. What are binomial coefficients, and how are they calculated?
2. How does the property $\sum_{i=0}^{n} \binom{n}{i} = 2^n$ arise from the binomial theorem?
3. How can binomial sums be used in combinatorics and probability?
4. What is the value of $2^{2023}$ approximately, and how can it be computed efficiently?
5. How does this problem connect to Pascal’s Triangle?

Tip:

When working with binomial coefficients, remember that symmetry can often simplify sums: $\binom{n}{i} = \binom{n}{n-i}$.