Answer to Question #98296 in Discrete Mathematics for Ahmed

Question #98296
Use mathematical induction to prove that 4 is a factor of 9n − 5
n
for all integers n ≥ 1.
1
Expert's answer
2019-11-11T11:04:34-0500

Putting n=1

We get

"9-5=4"

Which is divisible by 4 so 4 is the factor of 4


Let

"T(k) =9k-5k=4m\\ for \\ some \\ m" is true "\\ \\ \\ \\ ....(1)"


For

"T(k+1)"

Now,


"T(k+1)= 9(k+1)-5(k+1)"


"=9k+9-5k-5=4m+4"

"=4(m+1)"

Thus, "T(k+1)" is divisible by 4, hence 4 is a factor of it. By the principle of mathematical induction, 4 is a factor of 9n-5n for all integers "n \\geq 1."


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