Find the general term for the sequence whose first five terms are shown.
−1/8,1/27,−1/64,1/25,−1/216,...
To find the general term of a sequence, we need to observe the pattern within the given terms. Let's analyze the terms provided:
The sequence starts with -1/8, then continues with 1/27, -1/64, 1/25, -1/216.
Looking closely, we can see that the numerators (-1, 1, -1, 1, -1) alternate between -1 and 1.
Now, let's examine the denominators (8, 27, 64, 25, 216):
The denominators are not following a simple arithmetic or geometric pattern. However, upon observing, we notice that the denominators are perfect cubes: 2^3, 3^3, 4^3, 5^2, 6^3.
Hence, we can conclude that the general term of this sequence is given by:
(-1)^(n+1) / (n+1)^3
Where n is the position of the term in the sequence, starting with n = 1.