A chain of linked circular rings is made as shown at the rig

A chain of linked circular rings is made as shown at the right. Each ring has a thickness of 1 cm The fist ring has an outside diameter of 4 cm. Each ring moving down the chain has a diameter 1 cm longer than the previous ring. Complete the table of values for the total length of the chain from the top or the first ring to the bottom of the last ring. Explain how to find the length of chain if there are n rings in the chain. Clearly indicate your reasoning. If possible, find a model for the length of chain in terms of n. Adapted from American Mathematics Competition 12A, 2006

Solution

Reason :-

As we know the diameter of 1st ring is 4 cm

Then the diameter of 2nd ring = 4+1 = 5cm

Length of chain(With two rings) = diameter of first ring + (inner diameter of 2nd ring

Length of chain(With two rings) = 4 + (Outer diameter -2(thickness))

Length of chain(With two rings) = 4 + (5 -2(1)) = 4 +3 = 7

Length of chain(With three rings) = Length of chain(With two rings)+ (inner diameter of 3rd ring)

Length of chain(With three rings) = 7+ (Outer diameter -2(thickness))

(n=3) Length of chain(With three rings) = 7+ (6-2(1)) = 7+4= 11

(n=4) Length of chain(With four rings) = 11+ (7-2(1)) = 11+5= 16

(n=5) Length of chain(With five rings) = 16+ (8-2(1)) = 16+6= 22

Now we can see a patern by 4, 7, 11 , 16, 22.....

So our length of chain with n-rings is the nth term of this series.

L= Length of chain with n-rings =[(4+5+6+7......n) -2(n-1)]

Now n=1 , L = 4-2*(1-1) = 4

Now n=2 ,L = (4+5)-2*(2-1) = 7

Now n=3 , L = (4+5+6)-2*(3-1) = 11

Now n=4 , L = (4+5+6+7)-2*(4-1) = 16

Now n=5 , L = (4+5+6+7+8)-2*(5-1) = 22

L= Length of chain with n-rings =[(4+5+6+7......n) -2(n-1)]