To solve the equation [tex]\( S = \frac{n(n+1)}{2} \)[/tex] for [tex]\( n \)[/tex], follow these steps:
1. Clear the fraction: Multiply both sides of the equation by 2 to remove the denominator:
[tex]\[
2S = n(n + 1)
\][/tex]
2. Rearrange into a standard quadratic form: Expand the right side of the equation:
[tex]\[
2S = n^2 + n
\][/tex]
3. Set the equation to zero: Move all terms to one side to get a standard quadratic equation:
[tex]\[
n^2 + n - 2S = 0
\][/tex]
4. Identify the coefficients: The quadratic equation has the form [tex]\( an^2 + bn + c = 0 \)[/tex]. For our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -2S \)[/tex].
5. Use the quadratic formula: The solutions for [tex]\( n \)[/tex] in a quadratic equation [tex]\( an^2 + bn + c = 0 \)[/tex] are given by:
[tex]\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the coefficients [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -2S \)[/tex] into the formula:
[tex]\[
n = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2S)}}{2(1)}
\][/tex]
Simplify inside the square root:
[tex]\[
n = \frac{-1 \pm \sqrt{1 + 8S}}{2}
\][/tex]
6. Simplify the solutions:
[tex]\[
n = \frac{-1 \pm \sqrt{8S + 1}}{2}
\][/tex]
Thus, the solutions for [tex]\( n \)[/tex] are:
[tex]\[
n = \frac{-1 + \sqrt{8S + 1}}{2}
\][/tex]
and
[tex]\[
n = \frac{-1 - \sqrt{8S + 1}}{2}
\][/tex]
These correspond to:
[tex]\[
n = \sqrt{8S + 1}/2 - 1/2
\][/tex]
and
[tex]\[
n = -\sqrt{8S + 1}/2 - 1/2
\][/tex]
Therefore, the solutions to the equation [tex]\( S = \frac{n(n+1)}{2} \)[/tex] for [tex]\( n \)[/tex] are:
[tex]\[
n = \sqrt{8S + 1}/2 - 1/2
\][/tex]
and
[tex]\[
n = -\sqrt{8S + 1}/2 - 1/2
\][/tex]
