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**Punch** $\displaystyle 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5.............,k......$ is a sequence where the number $\displaystyle k$ appears $\displaystyle k$ times successively. $\displaystyle (k=1, 2, 3, 4, 5,...)$ Find the $\displaystyle 1000th$ term of the sequence.

$\displaystyle S_k=1000$

$\displaystyle 1+2+3+4+5+...+n=1000$

$\displaystyle \frac{n}{2}[2+(n-1)]$

$\displaystyle \frac{n}{2}[n+1]=1000$

$\displaystyle n(n+1)=2000$

$\displaystyle n^2+n-2000=0$

$\displaystyle n=-45.22(reject) or 44.22$

should i accept n=44 or n=45 as the answer?