Put the following statements into order to prove that is divisible by 3 for all non-negative integers n. Put N next to the statements that should not be used.
Question
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N 2 3 N N 5 4 N N 1 next to the statements that should not be used.What is sequence and example?
- A sequence is an ordered list of numbers . The three dots mean to continue forward in the pattern established.
- Each number in the sequence is called a term. In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on.
sequence- The statement is true for n = 0 because 0 is divisible by 3.
- Now suppose we have proved the statement for some n element of N This means that 4^n -1 = 3k for some integer k..
- We now consider the quantity 4^n+1 -1 = 4 . 4^n – 1 = 3 . 4^n + 4^n – 1
- We use the inductive hypothesis to simplify that equation to 4^n+1 – 1 = 3 . 4^n + 3k = 3 (4^n + k).
- We have thus shown that 4^n+ 1- 1 is a multiple of 3.
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