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The formula used for distance in $n$th second is $s=u+a/2(2n-1)$. In almost all the derivations it was formulated by subtracting the distance travelled in $(n-1)$ seconds from the distance travelled in $n$ seconds. Why not calculated by subtracting distance travelled in $n$ seconds from the distance travelled in $(n+1)$ seconds? I found the two formulas to be different. But what could be the reason for not formulating that way? Please let me know if I'm wrong.

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Think about the 1st second. By definition the distance travelled during the 1st second is equal to the distance attained after 1 second less the distance after 0 seconds.

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This is maybe more a question of the English language rather than Physics. The first second begins at $t = 0$ and ends at $t = 1$. Extrapolating, the nth second begins at $t = n - 1$ and ends at $t = n$.

For example, a baby's first year begins when he or she is born and ends on his or her first birthday.

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