Question #36578

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Answer 1 · 2017-10-27T18:41:49

See the proof below

We have to prove that #P(n)=(7^n-3^n)# is divisible by #4#

We prove this by mathematical induction

#n in NN#

When #P(0)=0#, #=>#, #0# is divisible by #4#, so the statement is true for #P(0)#

Suppose that the statement #P(n)# is true for #k#

Therefore,

#P(k)=7^k-3^k=4p#

#=>#, #3^k=7^k-4p#

Then , calculate #P(k+1)#

#P(k+1)=7^(k+1)-3^(k+1)#

#=7*7^k-3(7^k-4p)#

#=7*7^k-3*7^k+12p#

#=4*(7^k+3p)#

Therefore,

#P(k+1)# is divisible by #4#

We can conclude that #(7^n-3^n)# is divisible by #4# for all values of #n#