Compare two growth models

4 The circumference round the trunk of a large tree is measured and found to be 5.00 m . After one year the circumference is measured again and found to be 5.02 m .

1. Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 20 years after the first measurement.
2. Given instead that the circumferences at yearly intervals form a geometric progression, find the circumference 20 years after the first measurement.

8

1. The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
2. Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme \(A\) is to increase the amount of waste recycled each month by 0.16 tonnes.
   Scheme \(B\) is to increase the amount of waste recycled each month by \(6 \%\) of the amount recycled in the previous month.
   The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24 -month period. Scheme \(A\) Scheme \(B\) \(\_\_\_\_\)

1. Wheat is grown on a farm.

• In year 1 , the farm produced 300 tonnes of wheat.
• In year 12 , the farm is predicted to produce 4000 tonnes of wheat.

Model \(A\) assumes that the amount of wheat produced on the farm will increase by the same amount each year.

1. Using model \(A\), find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model \(B\) assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
2. Using model \(B\), find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
3. Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.

7. Kim starts working for a company.

• In year 1 her annual salary will be \(\pounds 16200\)
• In year 10 her annual salary is predicted to be \(\pounds 31500\)

Model \(A\) assumes that her annual salary will increase by the same amount each year.

1. According to model \(A\), determine Kim's annual salary in year 2 . Model \(B\) assumes that her annual salary will increase by the same percentage each year.
2. According to model \(B\), determine Kim's annual salary in year 2 . Give your answer to the nearest \(\pounds 10\)
3. Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10

6 An arithmetic progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 1.5\) for \(n \geqslant 1\).

1. Given that \(u _ { k } = 140\), find the value of \(k\). A geometric progression \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { n } = 120 \times ( 0.9 ) ^ { n - 1 }\) for \(n \geqslant 1\).
2. Find the sum of the first 16 terms of this geometric progression, giving your answer correct to 3 significant figures.
3. Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { N } u _ { n } > \sum _ { n = 1 } ^ { \infty } w _ { n }\).

12 Jim and Mary are each planning monthly repayments for money they want to borrow.

1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
   (A) Calculate his 12th payment.
   (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
   (A) Calculate her 12th payment.
   (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
   (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
   (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?

A small company which makes batteries for electric cars has a 10 year plan for growth. In year 1 the company will make 2600 batteries. In year 10 the company aims to make 12000 batteries. In order to calculate the number of batteries it will need to make each year from year 2 to year 9, the company considers two models. Model A assumes that the number of batteries it will make each year will increase by the same number each year.

1. According to model A, determine the number of batteries the company will make in year 2. Give your answer to the nearest whole number of batteries. [3]

Model B assumes that the numbers of batteries it will make each year will increase by the same percentage each year.

1. According to model B, determine the number of batteries the company will make in year 2. Give your answer to the nearest 10 batteries. [3]

Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.

1. Calculate the difference between the two totals, giving your answer to the nearest 100 batteries. [3]