To determine the value of [tex]\( a_n \)[/tex] given the formula [tex]\( a_n = \frac{(-1)^n}{2n-1} \)[/tex], we need to follow a few steps systematically. Let's break it down:
1. Understand the Expression:
- The given formula for [tex]\( a_n \)[/tex] is [tex]\( \frac{(-1)^n}{2n-1} \)[/tex].
- Here, [tex]\( n \)[/tex] is an integer that we will substitute into the formula to compute [tex]\( a_n \)[/tex].
2. Components of the Formula:
- [tex]\( (-1)^n \)[/tex] means that if [tex]\( n \)[/tex] is an even number, [tex]\( (-1)^n = 1 \)[/tex]; and if [tex]\( n \)[/tex] is an odd number, [tex]\( (-1)^n = -1 \)[/tex].
- [tex]\( 2n-1 \)[/tex] is an arithmetic operation. This term will always provide an odd number for any integer [tex]\( n \)[/tex].
3. Substitute and Simplification:
- Substituting [tex]\( n \)[/tex] into the formula and simplifying it gives us the value of [tex]\( a_n \)[/tex].
### Examples
Let's compute [tex]\( a_n \)[/tex] for a few values of [tex]\( n \)[/tex] to see how it works:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[
a_1 = \frac{(-1)^1}{2 \cdot 1 - 1} = \frac{-1}{1} = -1
\][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[
a_2 = \frac{(-1)^2}{2 \cdot 2 - 1} = \frac{1}{3}
\][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[
a_3 = \frac{(-1)^3}{2 \cdot 3 - 1} = \frac{-1}{5}
\][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[
a_4 = \frac{(-1)^4}{2 \cdot 4 - 1} = \frac{1}{7}
\][/tex]
Thus, for any integer [tex]\( n \)[/tex], the value of [tex]\( a_n \)[/tex] follows the formula:
[tex]\[
a_n = \frac{(-1)^n}{2n-1}
\][/tex]
In summary:
- [tex]\( a_n \)[/tex] changes its sign based on whether [tex]\( n \)[/tex] is odd or even, due to the term [tex]\( (-1)^n \)[/tex].
- The denominator [tex]\( 2n-1 \)[/tex] always ensures an odd number.
This is how you can determine the value of [tex]\( a_n \)[/tex] for any integer [tex]\( n \)[/tex].
