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Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence defined by

$$a_{1}=1, a_{n}=a_{n-1}+4 ; n \geqslant 2$$

Show that $a_{n}=4 n-3$ for all positive integers $n>1$

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in this problem, we wanna prove Ah, this statement that ah a n could be found using the recursive statement and minus one plus four is also equal to four n minus three for all positive integers greater than one. So we got to do our base case. We're gonna say for n equals two because it's greater than one. Uh, a two would be, Ah, a one plus four which is equal Thio one plus four, which is five. And on the right side. We got four times two minus three, which is eight minus three, which is five. And since the left side and right side are equal, we have shown that Ah, a one is true. Now for the inductive step we assume a k it is true or you can say and equals K is true both are fine So if a k is true, that means we have a K minus. One plus four is equal to four K minus three. Then for a K plus one, we have the following statement We'll take a plus. One can be found using that recursive formula by saying a K plus four a k on equivalent statement is four K minus three plus four and, ah, doing a little bit of, uh, combined like terms. First, this is four K plus one and then doing a little bit of rearranging this his four times K plus one minus three. And this was what we were trying to prove. We have put K plus one in place of n Therefore, we can say a K plus one is true when a k is true and ah therefore a n is true for all n in the positive integers that are greater then what?

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