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For the given number sequences, find the common difference or ratio and calculate the next two terms in each sequence.

Mathematics

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Here are the solutions for the number sequences and the word problem:

Number Sequences

  1. Sequence: −2;−6;−10;−14-2; -6; -10; -14
  1. Find the common difference.

    $$
    d = -6 - (-2) = -4
    $$
    
  2. Calculate the next two terms.

    $$
    -14 + (-4) = -18
    $$
    $$
    -18 + (-4) = -22
    $$
    The next two terms are $\boxed{-18, -22}$.
    

    2. Sequence f: −1;6;13;20-1; 6; 13; 20

  3. Find the common difference.

    $$
    d = 6 - (-1) = 7
    $$
    
  4. Calculate the next two terms.

    $$
    20 + 7 = 27
    $$
    $$
    27 + 7 = 34
    $$
    The next two terms are $\boxed{27, 34}$.
    

    3. Sequence: 13;21;29;3713; 21; 29; 37

  5. Find the common difference.

    $$
    d = 21 - 13 = 8
    $$
    
  6. Calculate the next two terms.

    $$
    37 + 8 = 45
    $$
    $$
    45 + 8 = 53
    $$
    The next two terms are $\boxed{45, 53}$.
    

    4. Sequence h: 0;1;2;30; 1; 2; 3

  7. Find the common difference.

    $$
    d = 1 - 0 = 1
    $$
    
  8. Calculate the next two terms.

    $$
    3 + 1 = 4
    $$
    $$
    4 + 1 = 5
    $$
    The next two terms are $\boxed{4, 5}$.
    

    5. Sequence: 7;5;3;17; 5; 3; 1

  9. Find the common difference.

    $$
    d = 5 - 7 = -2
    $$
    
  10. Calculate the next two terms.

    $$
    1 + (-2) = -1
    $$
    $$
    -1 + (-2) = -3
    $$
    The next two terms are $\boxed{-1, -3}$.
    

    6. Sequence: 2;4;82; 4; 8

  11. Find the common ratio.

    $$
    r = \frac{4}{2} = 2
    $$
    
  12. Calculate the next two terms.

    $$
    8 \times 2 = 16
    $$
    $$
    16 \times 2 = 32
    $$
    The next two terms are $\boxed{16, 32}$.
    

    Word Problem

    Thapelo's earnings pattern:

    • Week 1: R25
    • Week 2: R25 + R5 = R30
    • Week 3: R30 + R10 = R40 (The increase from Week 2 to Week 3 is R10, which is R5 more than the previous increase of R5).
  13. Determine the increase for Week 4.

    The increases are R5, R10. Following this pattern, the next increase will be R15.

  14. Calculate earnings for Week 4.

    Week 4 earnings=Week 3 earnings+increase=R40+R15=R55\text{Week 4 earnings} = \text{Week 3 earnings} + \text{increase} = \text{R}40 + \text{R}15 = \text{R}55

    Thapelo will earn R55\boxed{\text{R}55} in the fourth week.

  15. Calculate the total amount earned over 4 weeks.

    Total earnings=R25+R30+R40+R55=R150\text{Total earnings} = \text{R}25 + \text{R}30 + \text{R}40 + \text{R}55 = \text{R}150

    The total amount Thapelo earns over 4 weeks is R150\boxed{\text{R}150}.

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