JEE Mains · Maths · STD 11 - 8. sequence and series
Let \({S_n} = 1 + q + {q^2} + ..... + {q^n}\) and \({T_n} = 1 + \left( {\frac{{q + 1}}{2}} \right) + {\left( {\frac{{q + 1}}{2}} \right)^2} + ...... + {\left( {\frac{{q + 1}}{2}} \right)^n}\) where \(q\) is a real number and \(q \ne 1\). If \({}^{101}{C_1} + {}^{101}{C_2}.{S_1} + ...... + {}^{101}{C_{101}}.{S_{100}} = \alpha\, {T_{100}}\) then \(\alpha \) is equal to
- A \(2^{99}\)
- B \(202\)
- C \(200\)
- D \(2^{100}\)
Answer & Solution
Correct Answer
(D) \(2^{100}\)
Step-by-step Solution
Detailed explanation
\(\sum\limits_{r = 1}^{101} {{\,^{101}}{C_r}{s_{r - 1}}} \) \( = \sum\limits_{r = 1}^{101} {{\,^{101}}{C_r}\frac{{{q^r} - 1}}{{q - 1}}} \) \( = \frac{1}{{q - 1}}\left( {\sum\limits_{r = 1}^{101} {{\,^{101}}{C_r}{q^r} - \sum\limits_{r = 1}^{101} {{\,^{101}}{C_r}} } } \right)\)…
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