Arithmetic and Geometric Sequence Word Problems

Transcription

1 Name Date 6-11 Arithmetic and Geometric Series Word Problems Arithmetic and Geometric Sequence Word Problems How do you determine if a word problem is referring to an arithmetic sequence or a geometric sequence? 1. Look for keywords or clues in the question that tell you if the sequence is arithmetic or geometric. Arithmetic Sequence arithmetic, common difference, increases by a #, decreases by a #, etc.. Geometric Sequence geometric, common ratio, increases by a %, decreases by a %, etc.. 2. If they give you consecutive numbers (terms) try to see if there is a pattern to get from one term to the next Arithmetic Sequence adding or subtracting by a fixed number to get the next term Geometric Sequence multiplying or dividing by a fixed number to get the next term 1. Ann and John go to a swimming pool. They both swim the first length of the pool in 120 seconds. The time John takes to swim a length is 6 seconds more than he took to swim the previous length. The time Ann takes to swim a length is 1.05 times that she took to swim the previous length. (a) (i) Find the time John takes to swim the third length. (ii) Show that Ann takes seconds to swim the third length. (b) Find the time taken for Ann to swim a total of 10 lengths of the pool.

2 2. A National Lottery is offering prizes in a new competition. The winner may choose one of the following. Option one: Option two: Option three: $1000 each week for 10 weeks. $250 in the first week, $450 in the second week, $650 in the third week, increasing by $200 each week for a total of 10 weeks. $10 in the first week, $20 in the second week, $40 in the third week continuing to double for a total of 10 weeks. (a) Calculate the amount you receive in the tenth week, if you select (i) (ii) option two; option three. (b) What is the total amount you receive if you select option two? (c) Which option has the greatest total value? Justify your answer by showing all appropriate calculations.

3 3. Give all answers in this question correct to the nearest dollar Clara wants to buy some land. She can choose between two different payment options. Both options require her to pay for the land in 20 monthly installments. Option 1: Option 2: The first installment is $2500. Each installment is $200 more than the one before. The first installment is $2000. Each installment is 8 more than the one before. (a) If Clara chooses option 1, (i) (ii) write down the values of the second and third installments; calculate the value of the final installment; (iii) show that the total amount that Clara would pay for the land is $ (b) If Clara chooses option 2, (i) find the value of the second installment; (ii) show that the value of the fifth installment is $2721. (c) Clara knows that the total amount she would pay for the land is not the same for both options. She wants to spend the least amount of money. Find how much she will save by choosing the cheaper option.

4 Name Date 1. Rosa and Carlos both join a club in order to exercise Homework Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres. Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month. (a) (i) Write down the distance Rosa runs in the third training session; (ii) Write down the distance Rosa runs in the nth training session. (b) Calculate the total distance, in metres, Rosa runs in the first 50 training sessions. (c) Find the distance Carlos runs in the fifth month of training. (d) Calculate the total distance Carlos runs in the first year.

5 2. The first four terms of a sequence are 18, 54, 162, 486. (a) Use all four terms to show that this is a geometric sequence. (b) (i) Find an expression for the n th term of this geometric sequence. (ii) If the n th term of the sequence is , find the value of n. 3. In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term. (a) Find the common difference d. (b) Find the second term. (c) What is the sum of the first 90 terms of the sequence?

6 4. The tuition fees for the first three years of high school are given in the table below. Year These tuition fees form a geometric sequence. Tuition fees (in dollars) (a) (b) Find the common ratio, r, for this sequence. If fees continue to rise at the same rate, calculate (to the nearest dollar) the total cost of tuition fees for the first six years of high school. 5. Consider the following sequence: 997, 992, 987,..., 257, 252 (a) Find the common difference d. (b) Find the number of terms of the sequence. (c) Find the sum of the sequence.