Question

The power of \( x \) in the term with the greatest coefficient in the expansion of \( \left(1 + \frac{x}{2}\right)^{10} \) is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(5\)
• E. \(6\)

Hint:

Adjust for constants like \( \frac{1}{2} \) when finding greatest term in binomial expansion.

Answer 1

The Correct Option is C

Concept: Greatest term in binomial occurs near: \[ r = \frac{(n+1)|x|}{|1| + |x|} \]

Step 1: General term
\[ T_{r+1} = \binom{10}{r}\left(\frac{x}{2}\right)^r \]

Step 2: Max coefficient occurs at largest binomial coefficient
For \( (1+a)^n \), max coefficient near: \[ r = \frac{10}{2} = 5 \]

Step 3: Check effect of \( \frac{1}{2}^r \)
Coefficient decreases due to denominator So shift slightly left: \[ r = 4 \]

Step 4: Power of \(x\)
\[ x^r = x^4 \]

Step 5: Final answer
\[ \boxed{4} \]