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A sequence formula is a mathematical expression that defines the rule or pattern of a sequence. This rule is essential because it gives us a way to generate the terms of a sequence systematically. For any sequence, if you know the formula, you can find any term without listing all previous terms. Let's break down what a sequence formula consists of, using our example sequence where the formula is given by:\[ g(n)=\frac{3^n-1}{n^2} \]- **Variables and Constants:** In the formula above, you have both numbers and variables. The number 3 is a base of the exponent, indicating the growth pattern. Meanwhile, \(n\) is the variable that changes for each term you want to calculate.- **Exponents and Operations:** The term \(3^n\) indicates an exponential growth and the subtraction of 1 adjusts that growth for the sequence rule. The divisor \(n^2\) scales the result, impacting how quickly terms grow.Understanding this formula makes calculating any term straightforward. You simply plug in the appropriate value of \(n\) and follow the arithmetic operations as dictated by the formula.

Term calculation refers to determining individual elements of the sequence using the sequence formula. Using our example, you would calculate each term of the sequence by substituting different values of \(n\) into the sequence formula \(g(n)=\frac{3^n-1}{n^2}\). Let’s see how it happens step-by-step:

• **First Term**: For \(n=1\), substitute 1 into the sequence formula:\[ g(1) = \frac{3^1 - 1}{1^2} = 2 \]
• **Second Term**: For \(n=2\), substitute 2:\[ g(2) = \frac{3^2 - 1}{2^2} = \frac{8}{4} = 2 \]
• **Third Term**: For \(n=3\), substitute 3:\[ g(3) = \frac{3^3 - 1}{3^2} = \frac{26}{9} \]
• **Fourth Term**: For \(n=4\), substitute 4:\[ g(4) = \frac{3^4 - 1}{4^2} = 5 \]
• **Fifth Term**: For \(n=5\), substitute 5:\[ g(5) = \frac{3^5 - 1}{5^2} = \frac{242}{25} \]

Each calculation follows the same structure: replace \(n\) with the term number and perform the arithmetic as per the formula. It’s as simple as following a recipe, where you substitute \(n\) to get each new term.

In mathematics, sequences and series are fundamental concepts that describe collections of numbers arranged in a specific order. Unlike a set, where the order doesn't matter, in a sequence the order is crucial. Each number in a sequence is called a term, and there's a specific formula to find each term.- **Sequence**: A list of numbers in which each number is defined by a formula. For instance, \( g(n)=\frac{3^n-1}{n^2} \) generates a sequence where each \(n\) gives us a new number upon substitution.- **Series**: A series is related to a sequence and involves summing its terms. If you have the sequence 2, 2, \(\frac{26}{9}\), 5, \(\frac{242}{25}\), then the series would involve adding these numbers together.Both sequences and series are foundational for calculus and various mathematical applications such as finance and physics. Whether determining interest over time or waves in physics, these concepts help to model and solve real-world problems. Understanding how to manipulate and interpret sequences like the one in our example broadens our mathematical capability to predict and analyze trends and patterns.