Let S_n = 1+q+q^2+cdots+q^n and T_n = 1+dfrac q+1 2 +left(dfrac q+1 2 right)^2+cdots+left(dfrac q+1 2 right)^n, where q in mathbb R , qneq 1. If ^ 101 C_1 + ^ 101 C_2cdot S_1 + cdots + ^ 101 C_ 101 cdot S_ 100 = alpha, T_ 100 , find alpha.

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EEJ MAIN Mathematics QUESTION #6939

Question 1

Let $S_n = 1+q+q^2+\cdots+q^n$ and $T_n = 1+\dfrac{q+1}{2}+\left(\dfrac{q+1}{2}\right)^2+\cdots+\left(\dfrac{q+1}{2}\right)^n$, where $q \in \mathbb{R},\ q\neq 1$. If ${}^{101}C_1 + {}^{101}C_2\cdot S_1 + \cdots + {}^{101}C_{101}\cdot S_{100} = \alpha\, T_{100}$, find $\alpha$.

$2^{99}$
202
200
$2^{100}$✔️

Correct Answer Explanation

$S_n = \dfrac{q^{n+1}-1}{q-1}$. The LHS sum involves $\sum_{k=1}^{101}\binom{101}{k}S_{k-1}$.

After substituting and applying binomial theorem: the sum simplifies using $\sum \binom{101}{k}\frac{q^k-1}{q-1}$.

The result evaluates to $\alpha T_{100}$ where $\alpha = 2^{100}$.

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