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, we need to know the first term (A1), the last term (A40), and the total number of terms (n).
Given that A3 = -2 and A12 = 23, we can find the common difference (d) using the formula for the nth term of an arithmetic progression: An = A1 + (n-1)d.
For A3 = -2:
-2 = A1 + 2d (equation 1)
For A12 = 23:
23 = A1 + 11d (equation 2)
Now we have a system of two equations with two variables (A1 and d). Solve the system to find the values of A1 and d.
Subtracting equation 1 from equation 2, we get:
23 - (-2) = A1 + 11d - (A1 + 2d)
25 = 9d
So, d = 25/9.
Substituting the value of d into equation 1:
-2 = A1 + 2 * (25/9)
-2 = A1 + 50/9
-2 - 50/9 = A1
-18/9 - 50/9 = A1
A1 = -68/9
Now that we have A1, A40, and the total number of terms (n = 40), we can calculate the sum of the arithmetic progression (A1 + A2 + ... + An) using the formula:
Sum = (n/2) * (A1 + An)
= (40/2) * (-68/9 + [A1 + (n-1)d])
Substituting the values, we get:
Sum = 20 * (-68/9 + [-68/9 + (39 * (25/9))])
= 20 * (-68/9 + [-68/9 + (975/9)])
= 20 * (-68/9 + [-68/9 + 975/9])
= 20 * (-68/9 + 907/9)
= 20 * (839/9)
= 20 * 93.22
= 1864.4
Therefore, the sum of A1 + A2 + ... + A40 is 1864.4.
(b)
(i) According to Malthus' hypothesis, the population of England (Pn) would increase by 2% every year. This implies that:
Pn = Po * (1.02)^n
Where Po represents the population in 1800 and Pn is the population in 1800+n.
(ii) According to Malthus' hypothesis, the number of people that the English agriculture production can feed (An) will improve by an arithmetic progression, increasing by 400,000 people every year. This implies that:
An = Ao + 400,000n
Where Ao represents the number of people that the English agriculture production could feed in 1800, and An is the number of people it can feed in 1800+n.
(iii) To calculate the population of England in 1900 (P1900), we substitute n = 1900 into the formula from part (i):
P1900 = Po * (1.02)^1900
To calculate the number of people that the English agriculture production can feed in 1900 (A1900), we substitute n = 1900 into the formula from part (ii):
A1900 = Ao + 400,000 * 1900
(iv) To determine the year from which the English agriculture can no longer feed the English population according to Malthus' hypothesis, we can use a calculator to compare the increases in population (Pn) and the number of people that the English agriculture production can feed (An) by graphing or creating lists.
For example, you can create three lists on your calculator:
- L1 for n,
- L2 for Pn, and
- L3 for An.
Start with n = 1 and calculate Pn and An until the point where An is no longer greater than Pn. The corresponding year (n value) will then represent the year from which the English agriculture can no longer feed the English population.