# In which year do profits exceed 200,000 in the following case?A trading company made a profit of 50000 in year 1. The future profits are predicted to increase in a geometric series with common...

In which year do profits exceed 200,000 in the following case?

A trading company made a profit of 50000 in year 1. The future profits are predicted to increase in a geometric series with common ration r, r>1. If profits in year n are expected to exceed 200,000 prove that n > (log4)/(logr) +1

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We are given that the company makes profits of 50,000 in the first year and they are predicted to increase every year in such a manner that a geometric series is formed. As the common ratio is r, the profits in year 2 are 50,000*r, in year 3 they are 50,000*r^2 and so on. In year n the profits are 50000*r^(n - 1)

If profits exceed 200,000 in the year n, we have

50000*r^(n - 1) > 200000

divide both sides by 50000

=> r^(n - 1) > 4

take the log of both the sides

=> (n - 1)* log r > log 4

=> n - 1 > (log 4)/(log r)

=> n > [(log 4)/(log r)] + 1

**This gives the year n in which profits exceed 200000, one where n > [(log 4)/(log r)] + 1**