91Ó°ÊÓ

In algebraic sequences, the general term is a formula that defines how to calculate each term in the sequence. The general term is especially helpful because it provides a direct way to obtain any term in the sequence without listing all previous terms.
The formula given in this exercise is \( a_n = 3^{n-2} \). This means for any term number \( n \), the term \( a_n \) is determined by raising 3 to the power of \( n-2 \). With this formula, you can compute any term quickly.
To understand the general term, make sure you substitute different values for \( n \) to see the pattern of how the results change. This will help you become familiar with the concept of using a formula to find terms in a sequence.

• The "n" represents the term number in the sequence.
• The formula inside acts similar to a function, where input values provide outputs following a specific rule.

Calculating the terms of a sequence given a general term is all about substituting values. Once the formula is understood, the steps are straightforward:
1. Substitute the term number (n) into the formula.
2. Calculate the expression.

This method ensures you follow the pattern of the sequence accurately. Let's look at a recap of the calculations for each term in this specific sequence:
1. For \( n=1 \), the term is \( a_1 = 3^{-1} = \frac{1}{3} \).
2. For \( n=2 \), the term is \( a_2 = 3^{0} = 1 \).
3. For \( n=3 \), the term is \( a_3 = 3^{1} = 3 \).
4. For \( n=4 \), the term is \( a_4 = 3^{2} = 9 \).
5. For \( n=5 \), the term is \( a_5 = 3^{3} = 27 \).

By following these substitution steps for each value, you're performing a sequence calculation resulting in individual terms that make up the sequence.

• Make it easier by solving step by step.
• Ensure each substitution is correct so the pattern remains consistent.

In sequences, exponents play a crucial role in determining the value of each term. They are particularly common in sequences where growth or decay rates are detailed.
In the given equation \( a_n = 3^{n-2} \), the exponent \( n-2 \) determines the power to which the base (which is 3 in this exercise) is raised.
The importance of the exponent lies in how it controls the relationship between term positions and term values:

• Exponents affect the rate at which sequences grow; higher exponents typically mean larger numbers.
• Negative exponents, such as \( 3^{-1} \), result in fractions or values less than 1.

Understanding exponents allows one to predict how the sequence will behave as \( n \) increases or decreases. For instance:
- Positive exponents (like \( 3^1 \) or \( 3^2 \)) indicate that the sequence value grows larger.
- An exponent of zero (\( 3^0 \)) always results in 1.
Get comfortable with these variations and their outcomes to master sequences with exponents.