Question

The value \( \Gamma\!\left(\dfrac{5}{2}\right) \) is:

• A. \( \dfrac{3}{4}\sqrt{\pi} \)
• B. \( \dfrac{3}{8}\sqrt{\pi} \)
• C. \( \dfrac{3}{2}\sqrt{\pi} \)
• D. \( \dfrac{\sqrt{\pi}}{2} \)

Hint:

Apply \( \Gamma(x+1)=x\Gamma(x) \) twice from \( \Gamma(1/2)=\sqrt\pi \): \( \Gamma(5/2)=\tfrac32\cdot\tfrac12\sqrt\pi=\tfrac34\sqrt\pi \).

Answer 1

The Correct Option is A

Step 1: Use the recurrence \( \Gamma(x+1)=x\,\Gamma(x) \) and the base value \( \Gamma\!\left(\tfrac12\right)=\sqrt{\pi} \).

Step 2: Step down from \( \tfrac52 \):
\[ \Gamma\!\left(\tfrac52\right) = \tfrac32\,\Gamma\!\left(\tfrac32\right). \]

Step 3: Evaluate \( \Gamma\!\left(\tfrac32\right) \):
\[ \Gamma\!\left(\tfrac32\right) = \tfrac12\,\Gamma\!\left(\tfrac12\right) = \tfrac12\sqrt{\pi}. \]

Step 4: Substitute back:
\[ \Gamma\!\left(\tfrac52\right) = \tfrac32\cdot\tfrac12\sqrt{\pi} = \frac{3}{4}\sqrt{\pi}. \]
\[ \boxed{\, \Gamma\!\left(\tfrac52\right) = \dfrac{3}{4}\sqrt{\pi} \,} \]