Question

There are n arithmetic means between 1 and 31 such that seventh mean:(n-1)th mean =5:9. Find n.

Answer 1

If there are n means, then the common difference is (31-1)/(n+1) = 30/(n+1)

The kth mean is 1+k*30/n

So, you want

[1+7*30/(n+1)]/[1+(n-1)*30/(n+1)] = 5/9
n = 14

Check:
1,3,5,7,9,11,13,<b>15</b>,17,19,21,23,25,<b>27</b>,29,31