The population of a country increases by 2% every year. If it increases `k` times in a century, then prove that `[k]=7,w h e r e[dot]` represents the greatest integer function.

Explanation

If the initial number of inhabitants of the given country as A, then after a year, the total population will amount to
`A+(A)/(100)2=(1+(1)/(50))A`
After two years, thepopulation will amount to `(1+(1)/(50))^(2)A`.
After 100 years, it will reach the total of `(1+(1)/(50))^(100)A,`i.e., it will have increased `{(1+(1)/(50))^(50)}^(2)` times.
Taking into account that `underset(ntooo)lim(1+(1)/(n))^(n)~~e`, we can approximately consider that `(1+(1)/(50))^(50)~~e`.
Hence, after 100 yeras, the population of the country will have incereased `e^(2)~~7.39` times.
Hence, `[k]=[7.39]=7`.