Distance formula for nth second motion

Question:

Derive an expression for the distance travelled by a body in n^th second of its motion

Solution:

Distance Formula for nth Second Motion:

Distance travelled during the n^th second of its motion obtained by subtracting distance travelled in (n-1) second from the distance travelled in n seconds.

$${S_n \ th=S_n-S_(n-1)}$$

$${S_n \ th ={un+\frac{1}{2}an^2} – {[u(n-1)+\frac{1}{2}a(n-1)^2]}}$$

$${=un+\frac{1}{2} a n^2-un+u-\frac{1}{2}a[n^2+1-2n]}$$

$${=un+\frac{1}{2}an^2-un+u-\frac{1}{2}an^2-\frac{a}{2}+an}$$

$${S_n \ th= u+\frac{a}{2}[2n-1]}$$