Question

Find an explicit formula for a sequence of the form a₁, a₂, a₃, with the initial terms given below.

0, − 1/2 , 2/3 , − 3/4 , 4/5 , − 5/6 , 6/7
aₙ = ____

Answer 1

The explicit formula for the sequence 0, − 1/2, 2/3, − 3/4, 4/5, − 5/6,... is given by an = (-1)n+1(n − 1)/n. This captures the pattern of the sequence, alternating in sign and the relationship between the numerators and denominators for each term.

Step-by-step explanation:

The given sequence is 0, − 1/2, 2/3, − 3/4, 4/5, − 5/6,.... The pattern suggests an alternation in sign and a sequence of fractions where the numerator is one less than the denominator.

For the nth term an, the numerator appears to be n−1, and the denominator is n. The sign of each term alternates with each successive term, which we can represent with (-1)n+1, as n starts from 1. Combining these observations, we obtain the explicit formula for the sequence:

an = (-1)n+1(n − 1)/n

This formula provides the value of the nth term in the sequence given any positive integer n.