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

Verified step by step guidance

1. Identify the general term of the sequence, which is given as $a_n=3^n$.
2. Understand that the sequence is defined by the formula $a\_n\; =\; 3^n$, where $n$ represents the term number starting from 1.
3. Calculate the first term by substituting $n\; =\; 1$ into the formula: $a\_1\; =\; 3^1$.
4. Calculate the second term by substituting $n\; =\; 2$: $a\_2\; =\; 3^2$.
5. Calculate the third and fourth terms by substituting $n\; =\; 3$ and $n\; =\; 4$ respectively: $a\_3\; =\; 3^3$ and $a\_4\; =\; 3^4$.

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule or formula. Each number in the sequence is called a term, and the position of a term is indicated by its index n. Understanding how to identify and write terms from a general formula is fundamental.

General Term of a Sequence

The general term (an) of a sequence is a formula that allows you to find any term based on its position n. For example, an = 3^n means the nth term is 3 raised to the power of n. This formula helps generate terms without listing all previous ones.

Exponents and Powers

Exponents indicate repeated multiplication of a base number. In 3^n, 3 is the base and n is the exponent, meaning multiply 3 by itself n times. Understanding how to calculate powers is essential to find terms in sequences defined by exponential expressions.