ધારો કે $(1+x)^{10}$ ના વિસ્તરણમાં $x^{ r }$ નો દ્વિપદ્દી સહગગણક $C _{ r }$ વડે દર્શાવાય છે. જો $\alpha, \beta \in R$ માટે, $C _{1}+3 \cdot 2 C _{2}+5 \cdot 3 C _{3}+\ldots 10$ પદો સુધી = $\frac{\alpha \times 2^{11}}{2^{\beta}-1}\left(C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots 10\right.$ પદો સુધી $)$, તો $\alpha+\beta$ ની કિમત ....... છે.
$(1+x)^{10}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots \ldots+C_{10} x^{10}$
Differentiating
$10(1+x)^{9}=C_{1}+2 C_{2} x+3 C_{3} x^{2}+\ldots+10 C_{10} x^{9}$
replace $x \rightarrow X ^{2}$
$10\left(1+x^{2}\right)^{9}=C_{1}+2 C_{2} x^{2}+3 C_{3} x^{4}+\ldots+10 C_{10} x^{18}$
$10 \cdot x\left(1+x^{2}\right)^{9}=C_{1} x+2 C_{2} x^{3}+3 C_{3} x^{5}+\ldots .+10 C_{10} x^{19}$
Differentiating
$10\left(\left(1+x^{2}\right)^{9} \cdot 1+x \cdot 9\left(1+x^{2}\right)^{8} 2 x\right)$
$=C_{1} x+2 C_{2} \cdot 3 x^{3}+3 \cdot 5 \cdot C_{3} x^{4}+\ldots .+10 \cdot 19 C_{10} x^{18}$
putting $x=1$
$10\left(2^{9}+18 \cdot 2^{8}\right)$
$= C _{1}+3 \cdot 2 \cdot C _{2}+5 \cdot 3 \cdot C _{3}+\ldots+19 \cdot 10 \cdot C _{10} $
$C _{1}+3 \cdot 2 \cdot C _{2}+\ldots \ldots+19 \cdot 10 \cdot C _{10}$
$=10 \cdot 2^{9} \cdot 10=100 \cdot 2^{9}$
$C _{0}+\frac{ C _{1}}{2}+\frac{ C _{2}}{3}+\ldots . .+\frac{ C _{9}}{11}+\frac{ C _{10}}{11}=\frac{2^{11}-1}{11}$
$10^{\text {th }} \text { term } 11^{\text {th }} \text { term }$
$C _{0}+\frac{ C _{1}}{2}+\frac{ C _{2}}{3}+\ldots .+\frac{ C _{9}}{11}=\frac{2^{11}-2}{11}$
Now, $100 \cdot 2^{9}=\frac{\alpha \cdot 2^{11}}{2^{\beta}-1}\left(\frac{2^{11}-2}{11}\right)$
Eqn. of form $y = k \left(2^{ x }-1\right)$.
It has infinite solutions even if we take $x, y \in N$.
જો ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, તો ${C_0} + 2{C_1} + 3{C_2} + .... + (n + 1){C_n}$ = . . .
જો $f(y) = 1 - (y - 1) + {(y - 1)^2} - {(y - 1)^{^3}} + ... - {(y - 1)^{17}},$ હોય તો $y^2$ નો સહગુણક મેળવો.
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
જો $(1 - 2x + 5x^2 - 10x^3) (1 + x)^n = 1 + a_1x + a_2x^2 + ....$ આપેલ હોય અને $a_1^2\,= 2a_2$ હોય તો $n$ ની કિમત મેળવો
$\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$ ની કિમંત મેળવો.