Evaluate the expression Start Fraction n left parenthesis 2 n plus 1 right parenthesis over n minus 1 End Fraction for n equals two thirds.

Question

(1 point)
Responses

seven thirds
Image with alt text: seven thirds

negative Start Fraction 14 over 3 End Fraction
Image with alt text: negative Start Fraction 14 over 3 End Fraction

Start Fraction 14 over 15 End Fraction
Image with alt text: Start Fraction 14 over 15 End Fraction

negative 8 thirds

Answer 1

To evaluate the expression \(\frac{n(2n + 1)}{n - 1}\) for \(n = \frac{2}{3}\), we first substitute \(\frac{2}{3}\) for \(n\).

1. Calculate \(2n + 1\): \[ 2n + 1 = 2\left(\frac{2}{3}\right) + 1 = \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \]

2. Calculate \(n(2n + 1)\): \[ n(2n + 1) = \frac{2}{3} \cdot \frac{7}{3} = \frac{14}{9} \]

3. Calculate \(n - 1\): \[ n - 1 = \frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3} \]

4. Substitute \(n(2n + 1)\) and \(n - 1\) into the expression: \[ \frac{n(2n + 1)}{n - 1} = \frac{\frac{14}{9}}{-\frac{1}{3}} \]

5. Simplify: \[ \frac{14}{9} \div -\frac{1}{3} = \frac{14}{9} \cdot -3 = \frac{14 \cdot -3}{9} = -\frac{42}{9} \]

6. Simplify \(-\frac{42}{9}\): \[ -\frac{42}{9} = -\frac{14}{3} \]

Thus, the value of the expression for \(n = \frac{2}{3}\) is \(\boxed{-\frac{14}{3}}\).