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[" 18.Two soap bubbles of radii "r_(1)" and "r_(2)" equal to "4cm],[" and "5cm" are touching each other over a common "],[" surface "S_(1)S_(2)" (shown in figure).Its radius will be: "],[2^(n)(1-n sqrt(3))],[=(2^(n))/(2)((-n^(1/3))/(1/3-1))],[=(2n^(1/3-1))/(2^(1/3-1))quad (root(4)(5^(10)))^(28+sqrt(2))=(20)/(2^(2n))]

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(1^(3)+2^(3)+...+n^(3))/(1+3+5+...+(2n-1))=((n+1)^( 2))/(4)

to prove (2^(n)+2^(n-1))/(2^(n+1)-2^(n)))=(3)/(2)(3^(-3)*6^(2)*sqrt(98))/(5^(2)*((1)/(25))^((1)/(3))*(15)^(-(4)/(3))*3^((1)/(3)))=28sqrt(2)

(1)/(2n^(2)-1)+(1)/(3(2n^(2)-1)^(3))+(1)/(5(2n^(2)-1)^(5))+....=

(1^(2))/(1.3)+(2^(2))/(3.5)+(3^(2))/(5.7)+.....+(n^(2))/ ((2n-1)(2n+1))=((n)(n+1))/((2(2n+1)))

(1^(4))/(1.3)+(2^(4))/(3.5)+(3^(4))/(5.7)+......+(n^(4)) /((2n-1)(2n+1))=(n(4n^(2)+6n+5))/(48)+(n)/(16(2n+1))

lim_(n to oo)[(1)/(sqrt(n^(2)-1^(2)))+(1)/(sqrt(n^(2)-2^(2)))+(1)/(sqrt(n^(2)-3^(2)))+...+(1)/(sqrt(n^(2)-(n-1)^(2)))]