Understanding the nth Term of a Geometric Sequence: 3, 9, 27, 81, 243

The sequence provided in the question, 3, 9, 27, 81, 243, is a geometric sequence, not an arithmetic sequence. While an arithmetic sequence involves adding a constant to each term to get the next term, a geometric sequence involves multiplying each term by a constant to get the next term. In this case, each term is multiplied by 3 to get the next term.

Let's explore the sequence in detail:

Arithmetic Sequence vs Geometric Sequence

For an arithmetic sequence, let's say the first term is a1 and the common difference is d. The nth term is given by the formula:

an a1 (n-1)d

For a geometric sequence, let's say the first term is a1 and the common ratio is r. The nth term is given by the formula:

an a1 times; r(n-1)

Thus, for the sequence 3, 9, 27, 81, 243, the first term a1 is 3 and the common ratio r is 3. We can express the nth term as:

an 3 times; 3(n-1)

Simplifying the Formula

When simplifying the formula, we can see that:

an 3n

Let's break it down further to understand how this formula works:

For n 1, a1 31 3 For n 2, a2 32 9 For n 3, a3 33 27 For n 4, a4 34 81 For n 5, a5 35 243

By extending this pattern, we can see that the nth term of the sequence is indeed:

an 3n

Applications of Geometric Sequences

Geometric sequences have wide applications in various fields such as finance, physics, and engineering. For instance, in finance, geometric sequences can be used to model compound interest, where the amount of money grows exponentially. In physics, geometric sequences can model phenomena like wave propagation or decay processes.

In conclusion, the nth term of the sequence 3, 9, 27, 81, 243 is 3n. This formula can be used to find any term in the sequence, provided you know the value of n.

Key Points

The sequence provided is a geometric sequence. The common ratio of the sequence is 3. The nth term of the sequence can be represented by the formula 3n. This formula can be used to find any term in the sequence.