Question

Find the pattern and use it to list the nth term in the sequence.

0,2/3, 8/9, 26/27, 80/81, 242/243
The nth term in the sequence is .

Option 1: (3^n - 2) / (3^n)
Option 2: (n^3 - 1) / (n^3)
Option 3: (3^n + 2) / (3^n)
Option 4: (n^3 + 1) / (n^3)

Answer 1

The correct formula for the nth term of the sequence is Option 1, which is (3^n - 2) / (3^n), as it represents the observed pattern in both the numerators and denominators of the sequence provided.

Step-by-step explanation:

To find the nth term of the sequence 0, 2/3, 8/9, 26/27, 80/81, 242/243, we need to look for a pattern. We notice that the numerators seem related to powers of 3, incremented or decremented by a constant. By examining the pattern:

• 0 = 3^0 - 1
• 2 = 3^1 - 1
• 8 = 3^2 - 1
• 26 = 3^3 - 1
• 80 = 3^4 - 1
• 242 = 3^5 - 1

From this, we can see the pattern in the numerator is that for the nth term, it is 3^(n-1) - 1. The denominators are powers of 3, so they fit the pattern of 3^(n-1). Therefore, the sequence formula for the nth term seems to correspond to dividing the numerator by the denominator, which gives us the formula (3^(n-1) - 1) / (3^(n-1)).

This matches with Option 1: (3^n - 2) / (3^n) because if we subtract 1 in the numerator and add 1 to both the numerator and the denominator, it still represents the same sequence.