AP EAMCET · Maths · Binomial Theorem
If \(n \geq 100\) and the coefficient of \(x^{100}\) in \(1+(1+x)+(1+x)^2+\cdots+(1+x)^n\) is \({ }^{201} C_{101}\), then \(n=\)
- A 100
- B 200
- C 101
- D 190
Answer & Solution
Correct Answer
(B) 200
Step-by-step Solution
Detailed explanation
\(1+(1+x)+\cdots+(1+x)^n = \frac{(1+x)^{n+1}-1}{x}\) Coeff. of \(x^{100}\) in \(\frac{(1+x)^{n+1}-1}{x}\) is coeff. of \(x^{101}\) in \((1+x)^{n+1}-1\). Coeff. of \(x^{101}\) in \((1+x)^{n+1}\) is \({^{n+1}C_{101}}\). \({^{n+1}C_{101}} = {^{201}C_{101}}\) \(n+1 = 201\)…
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