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

b i)

a = 8 000 000
r = 1.2
Pn = 8 000 000 x 1.2^(n-1)

b ii)
a = 8 000 000
d = 400 000
Pn = 8 000 000 + (n-1)400 000
= 7 600 000 + 400 000n

This should help you to do the rest!

(a) To find the sum of an arithmetic progression (A.P.), we can use the formula:

Sum = (n/2) * (A1 + An)

Where n is the number of terms and A1 and An are the first and last terms of the A.P., respectively.

In this case, we are given that A3 = -2 and A12 = 23. We need to find the sum of A1 + ... + A40.

First, we need to find the common difference (d) of the A.P.:
d = (A12 - A3) / (12 - 3) = (23 - (-2)) / 9 = 25 / 9

We can then find A1 using A3 and the common difference:
A1 = A3 - 2d = -2 - 2*(25/9) = -2 - (50/9) = -16/9

Now, we can calculate the sum using the formula:
n = 40
An = A1 + (n - 1) * d = (-16/9) + (39) * (25/9) = 881/9

Sum = (40/2) * (-16/9 + 881/9) = 20 * (865/9) = 17300/9

Therefore, the sum of A1 + ... + A40 is 17300/9.

(b) (i) According to Malthus' hypothesis, the population of England (Pn) in the year 1800 + n can be expressed as:
Pn = Po * (1 + r)^n

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

(ii) According to Malthus' hypothesis, the number of people that the English agriculture production can feed (An) in the year 1800 + n can be expressed as:
An = Ao + (n - 1) * d

Where Ao is the number of people that agriculture can feed in 1800 (10 million) and d is the increase in the number of people agriculture can feed each year (400,000).

(iii) To calculate the population of England in 1900, we substitute n = 1900 - 1800 = 100 into the formula in (i):
P1900 = 8 million * (1 + 0.02)^100

To calculate the number of people that agriculture can feed in 1900, we substitute n = 1900 - 1800 = 100 into the formula in (ii):
A1900 = 10 million + (100 - 1) * 400,000

(iv) To determine the year from which the English agriculture can no longer feed the English population, we need to compare the increases in population (Pn) and the number of people agriculture can feed (An) by creating lists of years (L1), population (L2), and number of people agriculture can feed (L3). We can then use a graph or table to find the year where An becomes less than Pn.