Question

(C1/C0) + (2C2 /C1) + ( 3C​3/ C2) +.... + nCn/Cn-1​= ? Pls solve using summation method. Thanks

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \sf{ \frac{ C_1}{C_0} + 2 \frac{ C_2}{C_1} + 3\frac{ C_3}{C_2} + ... + n\frac{ C_n}{C_{n - 1}}}$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{ 1. \: \: C_r = {}^{n} C_r}$

$\displaystyle \sf{2. \: \: \frac{ {}^{n} C_r}{ {}^{n} C_{r - 1}} = \frac{n - r + 1}{r} }$

$\displaystyle \sf{ \implies \frac{ C_r}{ C_{r - 1}} = \frac{n - r + 1}{r} }$

EVALUATION

$\displaystyle \sf{ \frac{ C_1}{C_0} + 2 \frac{ C_2}{C_1} + 3\frac{ C_3}{C_2} + ... + n\frac{ C_n}{C_{n - 1}}}$

$\displaystyle \sf{ = \frac{n}{1} + 2 \times \frac{ n - 1}{2} + 3 \times \frac{n - 2}{3} + ... + n \times \frac{n - (n - 1)}{n}}$

$\displaystyle \sf{ = n + (n - 1) + (n - 2) + ...... + 1}$

$\displaystyle \sf{ = 1 + 2 + ..... + (n - 2) + (n - 1) + n}$

$\displaystyle \sf{ = \frac{n(n + 1)}{2} }$

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Answer 2

SOLUTION

TO EVALUATE

$\displaystyle \sf{ \frac{ C_1}{C_0} + 2 \frac{ C_2}{C_1} + 3\frac{ C_3}{C_2} + ... + n\frac{ C_n}{C_{n - 1}}}$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{ 1. \: \: C_r = {}^{n} C_r}$

$\displaystyle \sf{2. \: \: \frac{ {}^{n} C_r}{ {}^{n} C_{r - 1}} = \frac{n - r + 1}{r} }$

$\displaystyle \sf{ \implies \frac{ C_r}{ C_{r - 1}} = \frac{n - r + 1}{r} }$

EVALUATION

$\displaystyle \sf{ \frac{ C_1}{C_0} + 2 \frac{ C_2}{C_1} + 3\frac{ C_3}{C_2} + ... + n\frac{ C_n}{C_{n - 1}}}$

$\displaystyle \sf{ = \frac{n}{1} + 2 \times \frac{ n - 1}{2} + 3 \times \frac{n - 2}{3} + ... + n \times \frac{n - (n - 1)}{n}}$

$\displaystyle \sf{ = n + (n - 1) + (n - 2) + ...... + 1}$

$\displaystyle \sf{ = 1 + 2 + ..... + (n - 2) + (n - 1) + n}$

$\displaystyle \sf{ = \frac{n(n + 1)}{2} }$

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