27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.


{1, 3, 9, 27, 81, ......}

Verified step by step guidance

1. Identify the pattern in the given sequence: {1, 3, 9, 27, 81, ...}. Notice how each term relates to the previous one by multiplication.
2. Recognize that the sequence is geometric because each term is obtained by multiplying the previous term by a constant ratio. Calculate the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{3}{1} \).
3. Recall the general formula for the nth term of a geometric sequence: \( a_n = a_1 \times r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
4. Substitute the known values into the formula: \( a_1 = 1 \) and \( r = 3 \), so the explicit formula becomes \( a_n = 1 \times 3^{n-1} \).
5. Simplify the formula if possible. In this case, the explicit formula for the nth term is \( a_n = 3^{n-1} \).

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers where each number is called a term. The nth term, denoted as aₙ, represents the value at position n in the sequence. Understanding how terms progress helps in identifying patterns and formulating explicit expressions.

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. For example, in the sequence {1, 3, 9, 27, 81, ...}, each term is multiplied by 3. Recognizing this pattern is key to finding the explicit formula.

Explicit Formula for nth Term

The explicit formula expresses the nth term directly in terms of n, without needing previous terms. For geometric sequences, it is generally aₙ = a₁ * r^(n-1), where a₁ is the first term and r is the common ratio. This formula allows quick calculation of any term.