r1+3+3^(2)+...+3^(n-1)=((3^(n)-1))/(2)

Using the principle of mathematical induction, prove that 1.3 + 2.3^(2) + 3.3^(2) + ... + n.3^(n) = ((2n-1)(3)^(n+1)+3)/(4) for all n in N .

Using the principle of mathematical induction, prove that 1.3 + 2.3^(2) + 3.3^(2) + ... + n.3^(n) = ((2n-1)(3)^(n+1)+3)/(4) for all n in N .

(2.3^(n+1)+7.3^(n-1))/(3^(n+1)-2((1)/(3))^(1-n))=

lim_ (n rarr oo) (1+ (1) / (2) + (1) / (2 ^ (2)) + (1) / (2 ^ (3)) + ...... (1) / (2 ^ (n))) / (1+ (1) / (3) + (1) / (3 ^ (2)) + (1) / (3 ^ (3)) ...... (1) / (3 ^ (n)))

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

1.3+2.3^(2)+3.3^(3)+............+n.3^(n)=((2n-1)3^(n+1)+3 )/(4)

1^(3)+2^(3)+3^(3)+.....+n^(3)=(n(n+1)^(2))/(4), n in N

Prove that : 1^(3)+2^(3)+3^(3)++n^(3)={(n(n+1))/(2)}^(2)

sum_(r=1)^(n)(2r-1)=x then lim_(n to oo) [(1^(3))/(x^(2))+(2^(3))/(x^(2))+(3^(3))/(x^(2))+ .......+(n^(3))/(x^(2))] =

Let Delta_r=|{:(r,2r-1,3r-2),(n/2,n-1,a),((n(n-1))/2,(n-1)^2,1/2(n-1)(3n-4)):}| Show that sum_(r=1)^(n-1)Delta_r=0