Prove that the formula 11⋅2+12⋅3+13⋅4+...+1n(n+1)=n(n+1) is true for all natural numbers n.
By using mathematical induction,
Let P(n) denote the statement 11⋅2+12⋅3+13⋅4+...+1n(n+1)=n(n+1).
Then, we need to show that P(1) is true. So,
11⋅2=1(1+1)12=12
Thus, we prove the first principle of the mathematical induction. More over, assuming that P(k) is true, then
11⋅2+12⋅3+13⋅4+...+1k(k+1)=k(k+1)
Now, by showing P(k+1), we have
11⋅2+12⋅3+13⋅4+...+1k(k+1)+1(k+1)[(k+1)+1]=k+1[(k+1)+1]11⋅2+12⋅3+13⋅4+...+1k(k+1)+1(k+1)(k+2)=k+1k+2
We start with the left side and use the induction hypothesis to obtain the right side of the equation:
=[11⋅2+12⋅3+13⋅4+...+1k(k+1)]+[1(k+1)(k+2)]Group the first k terms=kk+1+1(k+1)(k+2)Induction hypothesis=k(k+2)+1(k+1)(k+2)Get the LCD=k2+2k+1(k+1)(k+2)Expand the numerator=(k+1)2(k+1)(k+2)Factor=k+1k+2Simplify
Therefore, P(k+1) follows from P(k), and this completes the induction step.
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