2(a)Consider the arithmetic progression with terms A3=-2 and A12=23. Find the sum of A1+......+A40 (can do)

(b)In 1800 the population of England was 8 million. The economist Malthus (1766-1834) produced a hypothesis, suggesting:-that the population of england would increase, according to a G.P., by 2% per year
-that the english agriculture production,able to feed 10 million people in 1800,would improve according to an A.P. to feed an extra 400000 people every year

Po represents the english population in 1800 and Pn that population in the year 1800+n:
(i)express,according to Malthus' hypothesis Pn as a function n.

Ao represents the number of people that the english agriculture production can feed in 1800 and An that number in 1800+n:
(ii)express,according to Malthus' hypothesis,An as a function of n
(iii)Calculate the population of england in 1900 and the number of people that the english agriculture production can feed in 1900
(iv)Determine the year from which the english agriculture can no longer feed the english population according to Malthus' hypothesis(-use your calculator by graphing or creating the lists:n=L1;Pn=L2;An=L3 tp compare increases).

(a) To find the sum of the arithmetic progression A1 + ... + A40, we can use the formula for the sum of an arithmetic series:

Sn = (n/2)(A1 + An)

Given that A3 = -2 and A12 = 23, we can find the common difference (d) and the first term (A1).

Using the formula for the nth term of an arithmetic progression:

An = A1 + (n - 1)d

Substituting A3 = -2 and A12 = 23:

-2 = A1 + (3 - 1)d
==> -2 = A1 + 2d (Equation 1)

23 = A1 + (12 - 1)d
==> 23 = A1 + 11d (Equation 2)

Solving Equations 1 and 2 simultaneously, we can find the values of A1 and d.

Subtracting Equation 1 from Equation 2:

23 - (-2) = (A1 + 11d) - (A1 + 2d)
==> 25 = 9d

Dividing both sides by 9:

d = 25/9

Substituting this value of d into Equation 1:

-2 = A1 + 2(25/9)
==> -2 = A1 + 50/9

Multiplying through by 9 to eliminate the fraction:

-18 = 9A1 + 50
==> 9A1 = -68
==> A1 = -68/9

Now we can find the sum of A1 + ... + A40 using the formula Sn = (n/2)(A1 + An).

Substituting n = 40, A1 = -68/9, and An = A1 + (n - 1)d, we can calculate the sum.

An = -68/9 + (40 - 1)(25/9)
==> An = -68/9 + 39(25/9)
==> An = -68/9 + 975/9
==> An = 907/9

Using the formula Sn = (n/2)(A1 + An):

S40 = (40/2)(-68/9 + 907/9)
==> S40 = (20)(839/9)
==> S40 = 8390/9
==> S40 = 932.22 (rounded to two decimal places)

Therefore, the sum of A1 + ... + A40 is approximately 932.22.

(b) (i) According to Malthus' hypothesis, the population in the year 1800+n (Pn) can be expressed as a function of n:

Pn = Po * (1 + r)^n

where Po is the population in 1800 (8 million), r is the growth rate (2% or 0.02), and n is the number of years after 1800.

(ii) According to Malthus' hypothesis, the number of people that the English agriculture production can feed in 1800+n (An) can be expressed as a function of n:

An = Ao + (n - 1)*d

where Ao is the number of people that the English agriculture production can feed in 1800 (10 million), and d is the rate of increase in food production (400,000 people per year).

(iii) To calculate the population of England in 1900 (P1900), substitute n = 1900 - 1800 = 100 into the population function:

P1900 = Po * (1 + r)^100
P1900 = 8,000,000 * (1 + 0.02)^100
P1900 ≈ 8,000,000 * 2.208
P1900 ≈ 17,664,000 (rounded to the nearest thousand)

To calculate the number of people that the English agriculture production can feed in 1900 (A1900), substitute n = 1900 - 1800 = 100 into the food production function:

A1900 = Ao + (100 - 1)*d
A1900 = 10,000,000 + 99 * 400,000
A1900 = 10,000,000 + 39,600,000
A1900 ≈ 49,600,000

(iv) To determine the year from which the English agriculture can no longer feed the English population, we need to find the year (n) where An < Pn.

Using the given population values and food production values, we can calculate An for each year and compare it to Pn until An < Pn.

Using a calculator or creating lists, we can compare the increases in An and Pn. The year where An < Pn is the year from which the English agriculture can no longer feed the English population according to Malthus' hypothesis.

(a) To find the sum of A1 + ... + A40, we need to find the common difference (d) and the first term (A1) of the arithmetic progression. We are given A3 = -2 and A12 = 23.

To find the common difference (d):
d = A12 - A3
= 23 - (-2)
= 25

To find the first term (A1):
A1 = A3 - 2d
= -2 - 2(25)
= -52

Now, we can find the sum of A1 + ... + A40 using the formula for the sum of an arithmetic series:

Sn = (n/2)(A1 + An)
= (40/2)(-52 + A40)

To find A40, we can use the formula for the nth term of an arithmetic sequence:

An = A1 + (n-1)d

A40 = A1 + (40-1)d
= -52 + 39(25)
= 913

Substituting the values into the sum formula:

Sn = (40/2)(-52 + 913)
= 20(-52 + 913)
= 20(861)
= 17220

Therefore, the sum of A1 + ... + A40 is 17220.

(b) (i) According to Malthus' hypothesis, the population Pn can be expressed as a function of n, where n represents the number of years since 1800.

Pn = Po * (1 + r)^n

Here, Po represents the initial population in 1800, which is 8 million, and r represents the rate of population increase, which is 2%. Convert the rate to a decimal: r = 2% / 100 = 0.02.

So, Pn = 8 million * (1 + 0.02)^n

(ii) According to Malthus' hypothesis, the number of people that the English agriculture production can feed, An, can be expressed as a function of n, where n represents the number of years since 1800.

An = Ao + d * n

Here, Ao represents the initial number of people that the English agriculture production can feed in 1800, which is 10 million, and d represents the increase in people that can be fed each year, which is 400,000.

So, An = 10 million + 400,000 * n

(iii) To calculate the population of England in 1900 (n = 1900 - 1800 = 100), we can substitute the value into the population function:

P1900 = 8 million * (1 + 0.02)^100

To calculate the number of people that the English agriculture production can feed in 1900 (n = 1900 - 1800 = 100), we can substitute the value into the agriculture function:

A1900 = 10 million + 400,000 * 100

(iv) To determine the year from which the English agriculture can no longer feed the English population, you can compare the values of Pn and An for increasing values of n. Use your calculator to graph or create lists.

Create lists:
L1: n = {0, 1, 2, ...}
L2: Pn = Po * (1 + 0.02)^n
L3: An = Ao + 400,000 * n

Graph the functions Pn and An, and find the point where they intersect or compare the values of Pn and An in the lists to find the year when An becomes less than Pn.