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In mathematics, a **sequence formula** defines the pattern or rule that governs the formation of a sequence. A sequence is essentially an ordered list of numbers following a specific pattern. For the sequence given by the formula \( a_n = \left(1 + \frac{1}{n}\right)^2 \), each term changes based on the value of \( n \).
The formula involves raising a fraction – the expression within the parentheses – to the second power. This means each sequence term is the result of squaring an expression that slightly varies as \( n \) increases.

• The term formula helps identify the nature of the sequence.
• It involves both constant values and variables that determine each term’s position in the sequence.

Underneath the sequence formula lies a consistent progression followed by every term in the sequence. By understanding this progression, you can calculate any term without having to list out preceding terms.

**Calculating a specific term** in a sequence involves substituting the term number into the sequence formula. Here, we aim to calculate the 100th term, denoted as \( a_{100} \).
To find \( a_{100} \), simply replace \( n \) in the formula with 100. This substitution transforms the sequence formula into \( a_{100} = \left(1 + \frac{1}{100}\right)^2 \).

• Replace the variable \( n \) with the term position to find the specific term.
• Calculate the expression: first handle operations inside parentheses.

For this sequence, after substituting \( n = 100 \), compute \( 1 + \frac{1}{100} \), which simplifies to \( \frac{101}{100} \).
Finally, you raise this result to the power of 2 to complete the term calculation, \( a_{100} = \left(\frac{101}{100}\right)^2 \), leading to our next step.

**Simplifying fractions** is crucial in many mathematical calculations, especially when dealing with sequences. Once we find the expression \( a_{100} = \left(\frac{101}{100}\right)^2 \), this step involves simplifying the fraction to make it more comprehensible.
Raising a fraction to a power requires elevating both the numerator and the denominator separately. That gives us:

• Calculate \( 101^2 = 10201 \).
• Calculate \( 100^2 = 10000 \).
• Thus, \( a_{100} = \frac{10201}{10000} \).

To express \( a_{100} \) more understandably, convert \( \frac{10201}{10000} \) into a decimal: **1.0201**.
This equivalent decimal representation shows the value of the 100th term in a simpler form, revealing a more common understanding of the result.