When something depreciates and 15%, its value
is multiplied by 100%-15% or 85% or 0.85.

So the geometric sequence is 

After     
 this      The
Number    value    
  of       of
years     the     
have      lathe
passed     is 
----------------------
  0       £3000
  1       £2550 
  2       £2167.50
  3       £1842.375
  4       £1566.01875
  5       £1551.115938
  6       £1121.448547
  7       £ 961.7312648
  8       £ 817.4715751
  9       £ 694.8508388
 10       £ 590.623213
 11       £ 502.0297311

The lathe will be sold after 11 years.

That's the easiest way, but your teacher perhaps
wants you do it using the formula for the nth term
of a geometric series:

By formula, a1 = £3000, r = 0.85





Take logarithms of both sides (you may use
either base logarithms, natural or common))





Since log(0.85) is negative, we must reverse
the inequality symbol from < to > when we
divide both sides by log(0.85)






 
The lathe will be sold after 11 years.

Edwin