Motion in a Straight Line

Step 1: Find acceleration a.
We use the formula for velocity with uniform acceleration: v=u+at. Here, u=0 (because the body starts from rest), and after n seconds, the velocity is v. So:
v=an⟹a=‌

v

n

Step 2: Find displacement in the nth second (Sn).
The formula for how far an object moves during the nth second is:
Sn=u+‌

a

2

(2n−1)
Since u=0 : Sn=‌

a

2

(2n−1)
Step 3: Find displacement in the (n−1) th second (Sn−1).
Plug ( n−1 ) instead of n into the formula:
Sn−1‌=‌

a

2

(2(n−1)−1)
‌=‌

a

2

(2n−3)
Step 4: Add the displacements for the nth and (n−1) th seconds.
Sn+Sn−1=‌

a

2

(2n−1)+‌

a

2

(2n−3)
Combine the numbers in the brackets:
Sn+Sn−1=‌

a

2

[(2n−1)+(2n−3)]=‌

a

2

(4n−4)
Step 5: Substitute a=‌

v

n

from Step 1.
Sn+Sn−1=‌

1

2

⋅‌

v

n

⋅(4n−4)=‌

v

2n

(4n−4)
Simplify:
Sn+Sn−1=‌

v

2n

⋅4(n−1)=‌

2v

n

(n−1)