Prove that the string an=n^2-n is strictly monotonic.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To prove that the given string is strictly monotonic, we'll have to verify if the difference between 2 consecutive terms is strictly positive or negative.

Since we know the form of the general term, we can determine the form of the next  term.

a(n+1) = (n+1)^2 - (n+1)

a(n+1) = n^2 + 2n + 1 - n - 1

a(n+1) = n^2 + n

Now, we'll check if the difference between a(n+1) and an is strictly positive or negative. If so, then the string is absolutely monotonic.

a(n+1) - an = n^2 + n - n^2 + n = 2n > 0

Since n is a natural number and it is bigger than 1 (n indicates the position of any term in the string), then the result 2n is strictly positive.

Therefore, the string whose general term is an = n^2 - n, is strictly monotonic.