91Ó°ÊÓ

The given sequence is,$\left\{a_k\right\}$.

The sequence $\left\{a_k\right\}$is eventually monotonic sequence, if it either eventually increasing or eventually decreasing.

Let $\left\{a_k\right\}$be a sequence and $\mathrm{ℕ}>0$be such that for all $k>\mathrm{ℕ},a_{k+1}≤a_k$, then the sequence is eventually decreasing sequence.

Let $\left\{a_k\right\}$be a sequence and $\mathrm{â„•}>0$be such that for all $k>\mathrm{â„•}$,$a_{k+1}â‰\yen a_k$, then the sequence is eventually increasing sequence.

In both the cases the sequence$\left\{a_k\right\}$is eventually monotonic sequence.

The terms of the sequence can be found by substituting $n=1,2,3,...$

Hence, the terms of the sequence are $1-1^2,2-2^2,3-3^2,....$

The terms of the sequence are, $0,-2,-6$.

The sequence is eventually decreasing sequence as role="math" localid="1649240464231" $a_2≤a_1,a_3≤a_2,.....,a_{k+1}≤a_k$.

Hence, the sequence is eventually monotonic.

Consider the sequence $\frac{1}{2},\frac{2}{3},\frac{3}{4},....,\frac{n}{n+1},..$

The sequence is eventually increasing sequence as, $a_2â‰\yen a_1,a_3â‰\yen a_2,.....,a_{k+1}â‰\yen a_k$.

Hence, the sequence is eventually monotonic.