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4. (a) Difference table is a table which consists of the differences of the each element with previous the one

1st table: -5, -1, 9, 25, 47, x

2nd table: 4, 10, 16, 22, y

3rd table: 6, 6, 6, 6

From the third table we can assume that y = 22 + 6 = 28

From the second table we can assume that x = 47 + 28 = 75

From the first table we can assume that the next term of the sequence is 84 + 75 = 159

(b) 1st difference table: 0, 2, 4, 6, 8, x

2nd difference table: 2, 2, 2, 2, 2

From the second difference table we can assume that x = 8 + 2 = 10

From the first difference table we can assume that the next term of the sequence is 30 + 10 = 40

7. (a) 1st step - understanding the problem

There is a next problem: we know how much the phone and the speaker costs together. Also we know the dependance of the price of the phone and speaker. We have to find the price of the phone. At the beginning we have equation with two arguments

2nd step - devise a plan

Since we know the total price and the dependance of the prices, we can express the value of the speaker through the value of the phone. After that we will receive equation with one argument that can be easily solved

3rd step - carry out the plan

Let x - price of the phone, then x - 2000 - price of the speaker

Now we have next equation: $x + x - 2000 = 6500\to x = 4250$

4th step - look back

We got the answer that the phone costs 4250, which means the speaker costs 2250. In total we get exactly 6500, which means our solving is correct. Also we use all of the giving information to provide an answer.

(b) 1st step

There is a next problem: we have 12 questions, all of them can be answered either 'true' or 'false'. It is required to calculate how many possible different chains of 12 answers exists.

2st step

We can look at this problem the next way: we have 12 steps(each step is answering one question). On each step we can make one of the two possible decisions, which means that on the first step we can build 2 chains(true or false). On the second step we also can make one of the two steps, but considering the previous step we can build 2*2 different chains in total. Following this lodic, after the 12 step we have $2^{12}$ possible chains.

3rd step

$2^{12} = 5148$ possible chains of answers

4th step

The received result seemed logical correctly, also the main idea can be easily verified by simple count when there is a less chain lenght. We also use all of the given information properly and provided correct answer