The value of _r=50^100^100C_rr+1 is:

Question

The value of $\displaystyle\sum_{r=50}^{100}\frac{{}^{100}C_r}{r+1}$ is:

Accepted Answer

Using $\dfrac{{}^{100}C_r}{r+1}=\dfrac{1}{101}\,{}^{101}C_{r+1}$,

$$\sum_{r=50}^{100}\frac{{}^{100}C_r}{r+1} = \frac{1}{101}\sum_{r=50}^{100}{}^{101}C_{r+1} = \frac{1}{101}\sum_{k=51}^{101}{}^{101}C_k$$

By symmetry of binomial coefficients, $\displaystyle\sum_{k=51}^{101}{}^{101}C_k = \frac{2^{101}}{2}=2^{100}$ (the upper half of the full expansion of $(1+1)^{101}$). Hence

$$\sum_{r=50}^{100}\frac{{}^{100}C_r}{r+1} = \frac{2^{100}}{101}$$

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