Proposition Essay Samples and Topic Ideas
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proposition is true for n=1For n=k≥1, the equation becomes 11*2+12*3+13*4+…+1k*k+1=nk+1If n=k+1 then we have 11*2+12*3+13*4+…+1k*k+1+1k+1*k+2=kk+1+1k+1*k+2=k*k+2+1(k+1)*(k+2)=k2+2k+1(k+1)*(k+2)=(k+1)2(k+1)*(k+2)=(k+1)(k+2)=(k+1)(k+1+1)…(i)Since k+1=n, i becomes nn+1 proving that 11*2+12*3+13*4+…+1nn+1=nn+1 for all n≥1Question 8 b. i=1ni22i-1(2i+1)=n(n+1)22n+1When n=1 the RHS is =1(1+1)22*1+1=13 and the LHS is i22i-1(2i+1)=14+2i-2i-1=13The RHS=LHS, therefore, the equation is right for n=1Suppose that the equation holds for n=k, then we have i=1k+1i22i-1(2i+1)=i=1ki22i-1(2i+1)+(k+1)2(2k+1-1)(2k+1+1))Through induction hypothesis we...
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