Question

If the 17th and 18th term in the expansion of \((2 + x)^{50}\) are equal, then the value of \(x\) is equal to

• A. 1
• B. 2
• C. 4
• D. 6
• E. 8

Hint:

Ratio of consecutive terms helps avoid large calculations.

Answer 1

The Correct Option is A

Concept: General term in binomial expansion: \[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]

Step 1: Identify terms
17th term $\Rightarrow$ $r=16$ 18th term $\Rightarrow$ $r=17$

Step 2: Equate terms
\[ \binom{50}{16} 2^{34} x^{16} = \binom{50}{17} 2^{33} x^{17} \]

Step 3: Simplify
\[ \frac{\binom{50}{16}}{\binom{50}{17}} \cdot \frac{2}{x} = 1 \] \[ \frac{17}{34} \cdot \frac{2}{x} = 1 \Rightarrow \frac{1}{x} = 1 \] \[ x = 1 \] Final Conclusion:
Option (A)