Thread: Arthimetic Series 3

A number of interlocking rings each 1 cm thick are hanging from a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance from the top of the top ring and the bottom of the bottom ring?

Here is a picture of what this thing looks like:


So, this is my attempt:

t1 = 20 cm d = -1 cm tn = 3 cm n = ? Sn = ? (We are trying to figure out Sn?)

First, I try to solve for n:

tn = t1 + (n-1)d

3 = 20 + (n-1)-1

3 = 20 + (n+1)

-17 = n + 1

n = -18

Now, I try to solve for Sn and plugging in the tn I found:

Sn = n/2 (t1 + tn)

Sn = -18/2 (20 + 3)

Sn = -207

The number shouldn't be positive...

The answer is actually 173 cm... I'm not sure where I went wrong... :/

You are taking circumference whereas we have to consider diameter to find the distance from top to bottom

I'm not sure how to do this question... Could you elaborate more?

You are going on the right track. Firstly it is the diameter which is given so you are right. In calculation n cannot be negative: You have gone wrong in opening bracket.
3 = 20 + (n-1)(-1)
3 = 20 -n +1
That will give n = 18
Now the sum of these terms is also correct i.e., 207
Now out of total 18 rings we have double length ( overlapping 1 cm each so we lose 32 cm on that count and 1 cm at top and bottom. taht makes a total of 34 cm . So the total distance would be 207-34 = 173 cm