Question

The census of deer in a forest is done every year starting from 2000.it was found that the population of deer is given as − + − . where x is the number of years after 2000.the number of guards in the forest is given by x-1.calculate an algebraic expression for the ratio of number of deer to the number of guards.

Answer 1

Answer:

Marco is a collector of antique soda bottles. His collection currently contains 437 bottles. Every year, he budgets enough money to buy 32 new bottles. Can we determine how many bottles he will have in 5 years, and how long it will take for his collection to reach 1000 bottles?

While both of these questions you could probably solve without an equation or formal mathematics, we are going to formalize our approach to this problem to provide a means to answer more complicated questions.

Suppose that P­n represents the number, or population, of bottles Marco has after n years. So P­0 would represent the number of bottles now, P­1 would represent the number of bottles after 1 year, P­2 would represent the number of bottles after 2 years, and so on. We could describe how Marco’s bottle collection is changing using:

P­0 = 437

P­n = P­n-1 + 32

This is called a recursive relationship. A recursive relationship is a formula which relates the next value in a sequence to the previous values. Here, the number of bottles in year n can be found by adding 32 to the number of bottles in the previous year, P­n-1. Using this relationship, we could calculate:

P­1 = P­0 + 32 = 437 + 32 = 469

P­2 = P­1 + 32 = 469 + 32 = 501

P3 = P­2 + 32 = 501 + 32 = 533

P­4 = P­3 + 32 = 533 + 32 = 565

P­5 = P­4 + 32 = 565 + 32 = 597

We have answered the question of how many bottles Marco will have in 5 years. However, solving how long it will take for his collection to reach 1000 bottles would require a lot more calculations.

While recursive relationships are excellent for describing simply and cleanly how a quantity is changing, they are not convenient for making predictions or solving problems that stretch far into the future. For that, a closed or explicit form for the relationship is preferred. An explicit equation allows us to calculate P­n directly, without needing to know P­n-1. While you may already be able to guess the explicit equation, let us derive it from the recursive formula. We can do so by selectively not simplifying as we go:

P­1 = 437 + 32 = 437 + 1(32)

P­2 = P­1 + 32 = 437 + 32 + 32 = 437 + 2(32)

P3 = P­2 + 32 = (437 + 2(32)) + 32 = 437 + 3(32)

P­4 = P­3 + 32 = (437 + 3(32)) + 32 = 437 + 4(32)

You can probably see the pattern now, and generalize that

P­n = 437 + n(32) = 437 + 32n

Using this equation, we can calculate how many bottles he’ll have after 5 years:

P­5 = 437 + 32(5) = 437 + 160 = 597

We can now also solve for when the collection will reach 1000 bottles by substituting in 1000 for P­n and solving for n

1000 = 437 + 32n

563 = 32n

n = 563/32 = 17.59

So Marco will reach 1000 bottles in 18 years.

bottles

In the previous example, Marco’s collection grew by the same number of bottles every year. This constant change is the defining characteristic of linear growth. Plotting the values we calculated for Marco’s collection, we can see the values form a straight line, the shape of linear growth.

Linear Growth

If a quantity starts at size P­0 and grows by d every time period, then the quantity after n time periods can be determined using either of these relations:

Recursive form:

P­n = P­n-1 + d

Explicit form:

P­n = P­0 + d n

In this equation, d represents the common difference – the amount that the population changes each time n increases by 1