Prove by induction that for all integers n &gt;= 1 , 3 ^ (2n - 1) + 1 is divisible by 4.

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Prove by induction that for all integers n &amp;gt;= 1 , 3 ^ (2n - 1) + 1 is divisible by 4.

nick1337 · Accepted Answer

To prove that 32n−1+1 is divisible by 4 for all integers n≥1 using mathematical induction, we must show two things:1. The statement is true for n = 1.2. If the statement is true for some k≥1, then it must also be true for k+1.**Basis Step:**Let&#039;s first prove the statement for the smallest possible value of n, i.e., n=1.For n=1, we have 32(1)−1+1=31+1=4. Since 4 is divisible by 4, the statement is true for n=1.**Inductive Step:**Now, let&#039;s assume that the statement is true for some k≥1. That is,32k−1+1 is divisible by 4.We must prove that the statement is also true for k+1, i.e.,32(k+1)−1+1 is divisible by 4.To do so, we can use the fact that 32(k+1)−1=32k+1·31=3·(32k)·31.Using the inductive hypothesis, we know that 32k−1+1 is divisible by 4. That is,32k−1+1=4m for some integer m.Substituting this into the expression for 32(k+1)−1, we get:32(k+1)−1+1=3·(32k)·31+1=3·(32k−1+1)+2.Since 32k−1+1=4m, we can substitute it into the above expression and simplify:32(k+1)−1+1=3·(4m)+2=12m+2=2(6m+1).Since 6m+1 is an integer, we have shown that 32(k+1)−1+1 is divisible by 4.Therefore, by mathematical induction, we have proven that for all integers n≥1, 32n−1+1 is divisible by 4.

Prove by induction that for all integers n

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