91Ó°ÊÓ

When we discuss inequalities in mathematics, we are dealing with a statement that indicates a relationship between two values, showing that one is larger or smaller than the other. Inequalities use symbols such as \(<\), \(>\), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to). In this exercise, the inequality \(\sum_{k=1}^{n} 2^{k} \leq 62\) represents the requirement that the sum of the series must remain less than or equal to 62.

This leads us to compare two values directly. In our problem, after using the summation formula, we are left with \(2^{n+1} \leq 64\), which simplifies our understanding of the limits placed on \(n\). Here, we need to find the maximum value for \(n\) that keeps the inequality true. The process involves rearranging and transforming the inequality until you isolate \(n\). This is achieved by factoring out shared values or by simplifying the expressions using mathematical operations which reflect equivalent comparisons.

Exponents are a way of expressing repeated multiplications of a number. For example, \(2^{3}\) means 2 is multiplied by itself three times, resulting in 8. In the context of the geometric series presented in this problem, exponents play a key role since the series is shown as \(2^{k}\).

The exponential property allows us to handle large numbers efficiently. When solving \(2^{n+1} \leq 64\), recognizing that 64 can be expressed as \(2^{6}\) allows us to work with \(2^{n+1} \leq 2^{6}\). This equivalence is crucial in solving the geometric series because it directly links the inequality to finding \(n\).

Using exponent rules, like \(a^{m} \cdot a^{n} = a^{m+n}\), can help simplify calculations and allow comparisons between expressions with different bases or exponents.

The summation formula for a geometric series is an essential tool for evaluating the sum of terms that increase exponentially. The formula \(S_n = a \frac{r^n - 1}{r - 1}\) allows us to calculate the total sum of the series when you know the first term \(a\), and the common ratio \(r\).

In this exercise, we start with \(a = 2\) and \(r = 2\). Plugging these into the formula, we modify it to \(S_n = 2\frac{2^n - 1}{1}\), simplifying further to \(2^{n+1} - 2\).

Understanding this formula is crucial because it offers a straightforward method to find the series sum. Once the formula is applied, the task converts from working with many individual terms to a more direct calculation. This simplification is vital when dealing with inequalities as it neatly frames the problem in terms of \(n\), making it easier to find solutions analytically and without manual counting of terms.