8. A metal post is repeatedly hit in order to drive it into the ground.
Given that
- on the 1st hit, the post is driven 100 mm into the ground
- on the 2nd hit, the post is driven an additional 98 mm into the ground
- on the 3rd hit, the post is driven an additional 96 mm into the ground
- the additional distances the post travels on each subsequent hit form an arithmetic sequence
- show that the post is driven an additional 62 mm into the ground with the 20th hit.
- Find the total distance that the post has been driven into the ground after 20 hits.
Given that for each subsequent hit after the 20th hit
- the additional distances the post travels form a geometric sequence with common ratio \(r\)
- on the 22 nd hit, the post is driven an additional 60 mm into the ground
- find the value of \(r\), giving your answer to 3 decimal places.
After a total of \(N\) hits, the post will have been driven more than 3 m into the ground.
Find, showing all steps in your working, the smallest possible value of \(N\).