A specialized geothermal drill is used to penetrate a layer of hard rock. During the 1st minute of drilling, the drill penetrates 160 cm. During the 2nd minute, it penetrates an additional 157 cm. During the 3rd minute, it penetrates an additional 154 cm. The additional depths penetrated in subsequent minutes form an arithmetic sequence.
Show that the drill penetrates an additional 100 cm during the 21st minute.
Find the total depth the drill has penetrated after the 21st minute.
After the 21st minute, the additional depths penetrated in each minute form a geometric sequence with common ratio rrr. On the 23rd minute, the drill penetrates an additional 94 cm.
Find the value of rrr, giving your answer to 3 decimal places.
After a total of N N\,N minutes, the drill will have penetrated a total depth of more than 50 m.
Find, showing your working, the smallest possible value of NNN.