Question

Solve each problems using Polya's four-step problem-solving strategy:

1. In the complex number system, i^1 = i; i^2 = -1; i^3 = -i; i^4 = 1; i^5 = i... Find i^173.

2. A coffee shop is giving away a signature annual planner. In the mechanics, each customer has to collect 24 stickers to avail of the said planner, and customers can share stickers. At the end of the promo period, John had a the most number of stickers, more than enough to get the planner. Unfortunately, Beth and Carl did not have enough. This is what they did: First, John gave Beth and Carl as many stickers as each had. Then, Beth gave John and Carl as many stickers as they had. At the end, each of the three friends had exactly enough stickers to get a planner. How many stickers did each person have at the start?

3. Ana and Ella decided to play a game. They collected 25 stones and placed it in an urn. They are to take turns taking away 1, 2, or 3 stones from the urn. The person who will take the last stone loses. Ana took the first move. What are her chances of winning the game?

4. The set {0,1} forms the binary system. There are exactly two 1-digit binary numbers namely 0 and 1; there are four 2-digit binary numbers namely 00, 01, 10, and 11; and so on... How many n-digit binary numbers can be formed?

Answer 1

(1)Given i;=-1,7 =-i;i 1;f5 = i, .. Require to findi73 Step 1: Understand the problem We observe that the given number i is a complex number given and few powers of the number i are -1,i3 =-i;i* 2 That is Step2: Devise a plan: From the given powers of the number i, we have _ = i1, Step3: Devise a plan: Carry out the plan Now express the number 173 as of the form 4n or 4n 1 or 4n-1 or 4-2 4-2 =-1i**-1 = -i 4겼+1 = i,i* where n = 1,2,3,4,... Using the above, we have 173 4 (43)+1 4(43)+ And13 Clearly the power of i (= 173) is of the form 4n1 4(43)+1 173 Therefore, (Using:i .4x+1 1 _ Step4: Look Back From the given data, we observe that the values of the powerss ofi are either 1 or 1 or i or -i -4겼 +1 1: .4것-2 = 1.7 4개 as 1 4x-1 where = 1,2,3,4.. according -1,i = -i = Here 173 173 is of the form i**+ and so i - 7 Hence, verified