^n{C_1}sumlimits_{r = 0}^1 {^1{C_r}} { + ^n}{ | vedclass

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Question

$^n{C_1} \cdot {2^1} + {{\mkern 1mu} ^n}{C_2} \cdot {2^2} + {{\mkern 1mu} ^n}{C_3} \cdot {2^3} +  \ldots . + {{\mkern 1mu} ^n}{C_n} \cdot {2^n}$

Vedclass · Accepted Answer

$(3^n-1)$

Vedclass · Answer

$^n{C_1} \cdot {2^1} + {{\mkern 1mu} ^n}{C_2} \cdot {2^2} + {{\mkern 1mu} ^n}{C_3} \cdot {2^3} +  \ldots . + {{\mkern 1mu} ^n}{C_n} \cdot {2^n}$

${(1 + 2)^n} = {{\mkern 1mu} ^n}{C_0} + {{\mkern 1mu} ^n}{C_1} \cdot {2^1} + {{\mkern 1mu} ^n}{C_2} \cdot {2^2} +  \ldots . + {{\mkern 1mu} ^n}{C_n} \cdot {2^n}$

$\left( {{3^n} - 1} \right) = {\,^n}{C_1} \cdot {2^1} + {\,^n}{C_2} \cdot {2^2} +  \ldots  \ldots  + {\,^n}{C_n} \cdot {2^n}$

$^n{C_1}\sum\limits_{r = 0}^1 {^1{C_r}} { + ^n}{C_2}\left( {\sum\limits_{r = 0}^2 {^2{C_r}} } \right){ + ^n}{C_3}\left( {\sum\limits_{r = 0}^3 {^3{C_r}} } \right) + ......{ + ^n}{C_n}\left( {\sum\limits_{r = 0}^n {^n{C_r}} } \right)$ ની કિમત મેળવો

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$\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$ ની કિમંત મેળવો.

$\sum\limits_{n = 1}^\infty {\frac{{^n{C_0} + ...{ + ^n}{C_n}}}{{^n{P_n}}}} $ = . . .

$-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots \ldots$ $-15 .{ }^{15} C _{15}+{ }^{14} C _{1}+{ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ નું મૂલ્ય ........ છે.

જો ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .......... + {C_n}{x^n}$, તો $\frac{{{C_1}}}{{{C_0}}} + \frac{{2{C_2}}}{{{C_1}}} + \frac{{3{C_3}}}{{{C_2}}} + .... + \frac{{n{C_n}}}{{{C_{n - 1}}}} = $

જો $(1 -x + x^2)^n = a_0 + a_1x + a_2x^2 + ....... + a_{2n}x^{2n}$,હોય તો $a_0 + a_2 + a_4 +........+ a_{2n}$ ની કિમત મેળવો