Find the nth term of a sequence whose first several terms are -1/3,1/9.-1/27,1/81
To find the nth term of this sequence, we need to identify the pattern.
From the given terms, we can see that the numerator alternates between -1 and 1, while the denominator follows a pattern of powers of 3 (3^1, 3^2, 3^3, ...).
So, the nth term can be written as:
(-1)^(n-1) / 3^(n-1)
Let's test this formula:
For n = 1,
(-1)^(1-1) / 3^(1-1) = 1 / 1 = 1.
For n = 2,
(-1)^(2-1) / 3^(2-1) = -1 / 3.
For n = 3,
(-1)^(3-1) / 3^(3-1) = 1 / 9.
For n = 4,
(-1)^(4-1) / 3^(4-1) = -1 / 27.
The formula produces the correct terms. Therefore, the nth term of the sequence is (-1)^(n-1) / 3^(n-1).
To find the nth term of a sequence, we first need to identify the pattern in the given sequence. Looking at the terms, we can observe that the sign alternates between negative and positive, and the denominator of each term is increasing sequentially as powers of 3 (3^1, 3^2, 3^3, ...).
Let's break down the sequence into parts and identify the patterns:
1st term: -1/3 = -1/3^1
2nd term: 1/9 = 1/3^2
3rd term: -1/27 = -1/3^3
4th term: 1/81 = 1/3^4
From this analysis, we can see that the general pattern is (-1)^(n+1) / 3^n, where n represents the position of the term.
So, the formula for the nth term of this sequence is (-1)^(n+1) / 3^n.
For example, if we want to find the 5th term, we substitute n = 5 into the formula:
5th term = (-1)^(5+1) / 3^5
= (-1)^6 / 3^5
= 1 / 243
Therefore, the 5th term of the given sequence is 1/243.