Let Sn = 1 + q + q2 + ........... qn and Tn = 1 + where q is a real number and q 1. If 101C1 + 101C2·S1 + .................+101C101· S100 = α T100 , then a is equal to

Engineering

Mathematics

Application of Binomial Theorem and Summation Of Series

Question

Let S_n = 1 + q + q² + ........... qⁿ and T_n = 1 + $(\frac{q + 1}{2}) + {(\frac{q + 1}{2})}^{2} + . ................... + {(\frac{q + 1}{2})}^{n}$ where q is a real number and q $\neq$ 1. If ¹⁰¹C₁ + ¹⁰¹C₂·S₁ + .................+¹⁰¹C₁₀₁· S₁₀₀ = α T₁₀₀ , then a is equal to

202

2⁹⁹

200

2¹⁰⁰

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${\sum_{r = 1}^{101}}^{101} C_{r} S_{r - 1}$

$= {\sum_{r = 1}^{101}}^{101} C_{r} \frac{q^{r} - 1}{q - 1}$

$= \frac{1}{q - 1} ({\sum_{r = 1}^{101}}^{101} C_{r} q^{r} - {\sum_{r = 1}^{101}}^{101} C_{r})$

$= \frac{1}{q - 1} ({(1 + q)}^{101} - 1 - 2^{101} + 1)$

$= \frac{α}{2^{100}} (\frac{{(1 + q)}^{101} - 2^{101}}{q - 1})$

⇒ a = 2¹⁰⁰