Question

1/1*2+1/2*3+ - - - - +1/n(n+1)=n/n+1 prove the following induction​

Answer 1

Answer:

Step-by-step explanation:

Let us assume that

$\large{\bold{P(n):\frac{1}{1.2}+\frac{1}{2.3}+..............+\frac{1}{n(n+1)}=\frac{n}{n+1}}}\\\\Consider\;\;P(1)\\\\LHS\;\;\frac{1}{1.2}=\frac{1}{2}\\\\RHS\;\;\frac{1}{1+1}=\frac{1}{2}\\\\LHS=RHS\\\\Therefore \;\;P(1)\;is\;True\\\\Let\;us\;assume\;that\;P(k)\;is\;TRUE\\\\\frac{1}{1.2}+\frac{1}{2.3}+..............+\frac{1}{k(k+1)}=\frac{k}{k+1}\\\\Add\;\frac{1}{(k+1)(k+2)}\;\;on\;\;both\;\;sides$

$=>\frac{1}{1.2}+\frac{1}{2.3}+..............+\frac{1}{k(k+1)}+\frac{1}{(k+1)(k+2)}=\frac{k}{k+1}+\frac{1}{(k+1)(k+2)}\\\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{1}{(k+1)(k+2)}+\frac{k}{k+1}\\\\=\frac{1}{k+1}(\frac{1}{k+2}+k)\\\\=\frac{1}{(k+1)(k+2)}(1+k(k+2))\\\\=\frac{1}{(k+1)(k+2)}(k+1)^2\\\\=\frac{k+1}{k+2}\\\\=\frac{(k+1)}{(k+1)+1}\\\\Therefore \;\;P(k+1)\;\;is \;\;True$

Therefore By Principle of Mathematical Induction P(n) is TRUE for all n∈N.

Hope it Helps