In Exercises 1–12, write the first four terms of each sequence whose general term is given. an=3^n

Verified step by step guidance

1. Identify the general term of the sequence, which is given as \( a_n = 3^n \).
2. To find the first term \( a_1 \), substitute \( n = 1 \) into the general term: \( a_1 = 3^1 \).
3. To find the second term \( a_2 \), substitute \( n = 2 \) into the general term: \( a_2 = 3^2 \).
4. To find the third term \( a_3 \), substitute \( n = 3 \) into the general term: \( a_3 = 3^3 \).
5. To find the fourth term \( a_4 \), substitute \( n = 4 \) into the general term: \( a_4 = 3^4 \).

Key Concepts

Sequences

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the position of each term is typically denoted by an index, such as 'n'. Understanding sequences is fundamental in algebra as they can represent various mathematical phenomena and are often used in functions and series.

Exponential Functions

Exponential functions are mathematical functions of the form f(n) = a * b^n, where 'a' is a constant, 'b' is the base, and 'n' is the exponent. In the given sequence, the general term an = 3^n represents an exponential function where the base is 3. These functions grow rapidly and are essential in modeling growth processes in various fields.

Evaluating Terms

Evaluating terms in a sequence involves substituting the index value into the general term formula to find specific terms. For the sequence an = 3^n, to find the first four terms, you would substitute n = 1, 2, 3, and 4 into the formula. This process is crucial for understanding how sequences develop and for calculating specific values within them.