Question

how to do principle of mathematical induction sums

Answer 1

First, take the given statement as n
For eg, P(n)= $5^{n}$ - 5 divisible by 4
Now, find p(1)
i.e, P(1)= $5^{1}$ -5 divisible by 4
=> 0 is divisible by 4 
thus, P(1) is true.
Assume P(k) to be true,
i.e, assume $5^{k}$ -5 to be divisible by4
take $5^{k}$ - 5= 4m (4 x anything is divisible by 4)
$5^{k}$ = 4m +5
P(k+1)= $5^{k+1}$ - 5 divisible by 4
           = 5.$5^{k}$ -5 
           = 5. (4m+5) -5
            = 20m +25 -5
             = 20+ 20m
             = 4(5m +5) which is divisible by 4 
thus, P(k+1) is true
=> P(k) is true whenever P(k+1) is true, for all values n element of Z