Arithmetic Progressions

Many patterns in daily life grow by a fixed amount: monthly instalments that increase by the same sum, a taxi fare that rises by a fixed charge for every extra kilometre, or the number of seats in successive rows of an auditorium. Such lists of numbers are arithmetic progressions (APs). In this chapter you will learn that in an AP each term after the first is obtained by adding a fixed number, the common difference d, to the preceding term, and that d may be positive, negative or zero. You will derive and use the formula for the nth term, a_n = a + (n - 1)d, to find any term, to locate a given term, or to check whether a number belongs to an AP. You will also derive the sum of the first n terms, S_n = (n/2)[2a + (n - 1)d] = (n/2)(a + l), the idea the young Gauss used to add 1 to 100, and apply these formulae to savings, salaries, production and seating problems.

Detailed Worksheet: Arithmetic Progressions Section A - Definitions (10 marks) 1. Define an arithmetic progression and its common difference. Give one example each of an AP with positive, negative and zero common difference. (2 marks) 2. Write the general form of an AP with first term a and common difference d. Write the formula for its nth term. (2 marks) 3. State the two formulae for the sum of the first n terms of an AP and explain what each symbol stands for. (2 marks) 4. What is meant by a finite AP and an infinite AP? Give one example of each. (2 marks) 5. Explain how the nth term a_n can be found when the sum S_n of the first n terms is known. Write the relation. (2 marks) Section B - Calculations and Applications (15 marks) 6. Which term of the AP 21, 18, 15, ... is -81? Is 0 a term of this AP? Give a reason. (3 marks) 7. Find the sum of the first 22 terms of the AP 8, 3, -2, ... (3 marks) 8. How many two-digit numbers are divisible by 3? Find their sum. (3 marks) 9. Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000? (3 marks) 10. The sum of the first 7 terms of an AP is 49 and the sum of the first 17 terms is 289. Find the sum of the first n terms. (3 marks) Section C - Diagrams (10 marks) 11. Matchsticks are used to make a row of squares: 1 square needs 4 sticks, 2 squares need 7 sticks and 3 squares need 10 sticks. Draw the first four figures of the pattern, write the number of sticks as an AP and find the number of sticks needed for 50 squares. (4 marks) 12. 200 logs are stacked so that there are 20 logs in the bottom row, 19 in the next row, 18 in the row after that, and so on. Draw a rough sketch of the stack for the first four rows. In how many rows are the 200 logs placed and how many logs are in the top row? (3 marks) 13. Draw a spiral made of successive semicircles with centres alternately at A and B, starting with centre A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... Find the total length of the spiral made up of 13 consecutive semicircles. (Use pi = 22/7) (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are 10 potatoes. A competitor starts from the bucket, picks up the nearest potato, runs back to drop it in the bucket, and continues until all the potatoes are in the bucket. Show that the distances form an AP and find the total distance run. (5 marks) 15. How many terms of the AP 24, 21, 18, ... must be taken so that their sum is 78? You will get two answers. Explain why both answers are valid by examining the terms of the AP. (5 marks) 16. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the formula used and every step of working. Check each answer by substituting back where possible.