The nth term of the sequence 3, 9, 27, 81.....is

Analysis

The sequence provided is 3, 9, 27, 81, ... We need to find the general formula (the $n^{th}$ term) that generates these numbers.

Concept: Geometric Progression (GP)

A sequence is a geometric progression if the ratio between consecutive terms is constant. This constant is called the common ratio ($r$).

• The first term ($a$) = 3
• The second term = 9
• The third term = 27

To find the common ratio ($r$): $r = \frac{\text{second term}}{\text{first term}} = \frac{9}{3} = 3$

The formula for the $n^{th}$ term ($T_n$) of a geometric progression is: $T_n = a \cdot r^{(n-1)}$

Step-by-Step Solution

1. Identify the variables: • $a = 3$ • $r = 3$

2. Substitute these values into the formula: $T_n = 3 \cdot 3^{(n-1)}$

3. Verify the formula: • For $n = 1$: $T_1 = 3 \cdot 3^{(1-1)} = 3 \cdot 3^0 = 3 \cdot 1 = 3$ (Correct) • For $n = 2$: $T_2 = 3 \cdot 3^{(2-1)} = 3 \cdot 3^1 = 3 \cdot 3 = 9$ (Correct) • For $n = 3$: $T_3 = 3 \cdot 3^{(3-1)} = 3 \cdot 3^2 = 3 \cdot 9 = 27$ (Correct)

Comparing our derived formula $3 \cdot 3^{n-1}$ with the given options, it matches option B.

Conclusion

The $n^{th}$ term of the sequence is $3 \times 3^{n-1}$.

Exam Tip

When you see a sequence where each term is a power of the same number (like 3, 9, 27...), always check if it is a geometric progression. If the first term is the same as the common ratio, the formula will always simplify to $a^n$ or $r^n$. In this case, $3^1, 3^2, 3^3, 3^4...$ which is simply $3^n$. Since the options are written in the form $3 \times 3^{n-1}$, remember that $3^1 \times 3^{n-1} = 3^{1+n-1} = 3^n$.