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The calculation of terms in a sequence, especially a geometric sequence, follows a consistent pattern where each term can be found using a specific formula. In the case of a geometric sequence, the general term is given by the formula \(a_n = a \times r^n\), where \(a\) is the first term, and \(r\) is the common ratio, the constant factor between consecutive terms of the sequence.
In our example, we have the formula \(a_{n} = 5(3^{n})\), which represents a geometric sequence with the first term \(5\) and a common ratio of \(3\). Calculating the first few terms of a sequence is straightforward. For the first term, \(a_{0}\), we substitute \(n = 0\) into the formula to get \(a_{0} = 5(1) = 5\), since any non-zero number to the power of 0 is \(1\). Similarly, to calculate \(a_{1}\), we put \(n = 1\) and for \(a_{2}\), \(n = 2\), yielding \(a_{1} = 15\) and \(a_{2} = 45\), respectively.

• To calculate the \(n\)-th term of a geometric sequence, plug in the value of \(n\) into the predetermined formula.
• Understand the first term and the common ratio as they define the entire sequence's progression.
• For any term, the power of \(n\) correlates directly with its position in the sequence.

Exponential expressions are mathematical notations that involve raising numbers to a power, indicating repeated multiplication of a base number. They are written in the form of \(b^n\), where \(b\) is the base and \(n\) is the exponent or power, and they represent the base \(b\) multiplied by itself \(n\) times.
Geometric sequences often involve exponential expressions, as each term can be expressed as the product of a constant and the base raised to the nth power. Understanding how to manipulate these expressions is crucial for working with geometric sequences.

Simplifying Exponential Expressions

When simplifying exponential expressions like in our exercise, it is important to remember the following:

• Any number to the power of 0 is always 1, hence \(3^0 = 1\).
• When the exponent is a positive integer, simply multiply the base by itself the number of times indicated by the exponent. For example, \(3^2 = 3 \times 3 = 9\).
• The coefficient, in this case, \(5\), is multiplied by the result of the exponential expression to yield each term of the sequence.

Recognizing these properties allows for the efficient computation and simplification of the terms in a geometric sequence.

Mathematical Induction is a powerful proof technique used in mathematics to establish that a given property is true for all natural numbers. It follows a principle akin to dominos falling—showing that if the first one falls (base case) and if each domino causes the next to fall (inductive step), then all the dominos will fall.
In the context of sequences, especially geometric sequences, mathematical induction can be used to prove properties of the sequence, like formulas for the nth term, or to show that a certain property holds for all terms of the sequence. It involves two key steps:

• Base Case: Verify the property for the initial term of the sequence, often for \(n = 0\) or \(n = 1\).
• Inductive Step: Show that if the property holds for an arbitrary term \(a_k\), then it must also hold for the next term \(a_{k+1}\).

The beauty of mathematical induction lies in its logical simplicity and its powerful ability to confirm infinite cases with just two steps. It is a fundamental tool in understanding how geometric sequences operate and proving related conjectures within the realm of mathematics.