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When we say a sequence is "strictly increasing," we mean that each term in the sequence is greater than the one before it. This concept is like climbing a set of stairs, where each step is higher than the last. In mathematical terms, a sequence \( \{a_n\} \) is strictly increasing if for every \( n \), \( a_{n+1} > a_n \). This ensures there is a constant upward trend without any plateaus or dips.

For example, consider the sequence \( 1 - \frac{1}{n} \). To show it's strictly increasing, we look at the difference \( a_{n+1} - a_n \). If this difference is positive for all \( n \), then each term is truly larger than the last, confirming the sequence is strictly increasing.

• Climbing stairs analogy: each term is a step higher.
• Mathematical criterion: \( a_{n+1} - a_n > 0 \).
• A positive difference assures strict increase.

Understanding strictly increasing sequences is crucial because they represent upward trends. This can be particularly helpful in analyzing patterns where growth is expected.

The difference between consecutive terms in a sequence is a powerful tool to determine if the sequence is increasing or decreasing. This involves calculating \( a_{n+1} - a_n \), which gives the numerical gap between the subsequent term and the current term. If this gap - or difference - is positive, then the sequence is increasing. If negative, the sequence is decreasing.

In our sequence \( 1 - \frac{1}{n} \), we calculated the difference as \( \frac{1}{n} - \frac{1}{n+1} = \frac{1}{n(n+1)} \). Notice the positive result, \( \frac{1}{n(n+1)} \), which means each term surpasses the one before it, confirming a strictly increasing pattern.

• Difference calculation shows trends in the sequence.
• Positive difference indicates increase, negative indicates decrease.
• This calculation solidifies understanding of the sequence's behavior.

By breaking down the difference, you can visualize how each term interacts with the next. This method of comparison is especially useful in mathematical proofs and when predicting future terms.

The general term \( a_n \) offers a formula to calculate any term in the sequence, which is foundational in understanding the sequence's behavior. In our sequence, the general term is expressed as \( a_n = 1 - \frac{1}{n} \). This form provides a snapshot of how terms are generated based on their position \( n \) in the sequence.

The importance of the general term lies in its predictability; it helps you find any term without listing all previous terms. For instance, if you want to find the 100th term of the sequence, simply plug in \( n = 100 \) to get \( a_{100} = 1 - \frac{1}{100} \).

• General term allows calculation of any term directly.
• Offers insight into the sequence's formation and progression.
• Facilitates deeper understanding and is vital for analysis.

Using the general term not only aids in calculating specific terms but also helps in deriving overall properties of the sequence, like limits and convergence.