Question

Can someone please help me?

Use mathematical induction to prove the following.

Please show your work.

13.
`(1)/(1(2)) +(1)/(2(3))+(1)/(3(4)) +...+(1)/(n(n+1)) =(n)/(n+1)`


Any help is greatly appreciated!

Answer 1

im doing the same work but i am working on it ill help you afterwards when i am done .

Explanation:

Answer 2

First show it's true for
`n=1`. On the left,

`\frac1{1\cdot2}=\frac12`

On the right,

`\frac1{1+1}=\frac12`

so the base case
`n=1` is true.

Now assume equality holds for
`n=k`, that

`\frac1{1\cdot2}+\frac1{2\cdot3}+\cdots\frac1{k(k+1)}=\frac k{k+1}`

We use this assumption to show it also holds for
`n=k+1`. By hypothesis,

`\frac1{1\cdot2}+\frac1{2\cdot3}+\cdots+\frac1{k(k+1)}+\frac1{(k+1)(k+2)}=\frac k{k+1}+\frac1{(k+1)(k+2)}`

(the first
`k` terms condense to
`\frac1{k(k+1)}`)

Combining the fractions gives

`(k(k+2))/((k+1)(k+2))+\frac1{(k+1)(k+2)}=(k^2+2k+1)/((k+1)(k+2))=((k+1)^2)/((k+1)(k+2))=(k+1)/(k+2)`

which is what we had to establish, thus proving (by induction) equality for all
`n`.