Asked by RAMADAN MAHBUB MOHAMMED
Using Mathematical Induction, prove the following:
a) 6^n - 1 is divisible by 5, for n>_0.
a) 6^n - 1 is divisible by 5, for n>_0.
Answers
Answered by
oobleck
check P(0).
6^0-1 = 0, which is divisible by 5
Assume P(k). Now,
6^(n+1)-1 = (5+1)^(n+1) - 1
Now use the Binomial Theorem to expand that.
= 5^(n+1) + C(n,1)*5^n*1 + ... + C(n,n-1)*5*1^n + 1^(n+1) - 1
= 5(5^n + C(m,1)*5^(n-1) + ... + C(n,n-1))
which is divisible by 5.
6^0-1 = 0, which is divisible by 5
Assume P(k). Now,
6^(n+1)-1 = (5+1)^(n+1) - 1
Now use the Binomial Theorem to expand that.
= 5^(n+1) + C(n,1)*5^n*1 + ... + C(n,n-1)*5*1^n + 1^(n+1) - 1
= 5(5^n + C(m,1)*5^(n-1) + ... + C(n,n-1))
which is divisible by 5.
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