Question

At present the population of a village is p and if increase rate of population per year be 2r%, the population will be after n years

Answer 1

p(x) = p*(1 + 2r%)^n , where n is the years since the initial population, p.

or

p(x) = p*(1 + 0.02r)^n

Step-by-step explanation:

Let n be the number of years after the year the village has population p.

We can write:

p(0) = p [population is p for the initial year for which the population is known].

p is the population of the village at the start (n = 0) and it increases by 2r%/year. The population the next year, n+1, can be determined by:

p(1) = p*(1 + 2r%)

Since p increases by the same rate in following years, we can see that the term (1 + 2r%) would be multiplied again, for the number of additional years since n(0), the initial year. We can try a few examples:

p(0) = p

p(1) = p*(1 + 2r%)

p(2) = p*(1 + 2r%)*(1 + 2r%)

p(3) = p*(1 + 2r%)*(1 + 2r%)*(1 + 2r%)

p(4) = p*(1 + 2r%)*(1 + 2r%)*(1 + 2r%)*(1 + 2r%)

and so on.

Rather than wasting pixels writing out, say, n (12), we can write a general form that is valid for all years by noting that (1 + 2r%) is multiplied times itself for n times:

p(0) = p*(1 + 2r%)^0

p(1) = p*(1 + 2r%)^1

p(2) = p*(1 + 2r%)^2

p(3) = p*(1 + 2r%)^3*

p(4) = p*(1 + 2r%)^4

We can write this as

p(x) = p*(1 + 2r%)^n , where n is the years since the initial population, p.

or

p(x) = p*(1 + 0.02r)^n

Answer 2

The population of a village after n years can be found using the formula P = p(1 + 2r/100)^n.

Step-by-step explanation:

The population of a village at present is represented by p. If the increase rate of population per year is 2r%, then the population after n years can be found using the formula: P = p(1 + 2r/100)^n.

For example, if the current population of the village is 1000 and the increase rate is 3%, then the population after 5 years would be:

P = 1000(1 + 2(3)/100)^5

= 1000(1 + 0.06)^5

= 1000(1.06)^5

= 1000(1.338225)

≈ 1338.23.