4 Shen is trying to solve this problem. Work out the answer to the problem. Explain the method you used to solve the problem. The fifth term of a non-linear sequence is 9. The term-to-term rule is 'divide by 3'. Work out the second term of the sequence. 5 Use the position-to-term rule to work out the first four terms of each sequence. a term position number + 5 c term = position number² + 4 b term 3 x position number - 1 d term 3 x position number² = 6 Use the position-to-term rules to work out: i the fifth term ii the 10th term a term = 4 x position number + 1 iii the 20th term of each sequence. b term position number - 2 7 The third term of a sequence is 25. The fifth term of the sequence is 73. Which of these position-to-term rules is the correct one for the sequence? A term = position number² + 16 C term = 3 x position number - 2 Show how you worked out your answer. B term 5x position number + 10 D term 9x position number - 2 8 This is Shen's homework. He has spilt tea on some of the questions and answers! D Question I The position-to-term rule of a sequence is term=2x position number + Work out the first three terms of the sequence

Let's tackle 4 Shen's questions step by step.

Question 4: Non-linear sequence

Problem: The fifth term of a non-linear sequence is 9, and the term-to-term rule is "divide by 3." We need to find the second term of the sequence.

Solution: To find the second term, we will work backward from the fifth term.

1. Fifth term (T5) = 9
2. To find the fourth term (T4), we reverse the operation:
   • T4 = T5 * 3
   • T4 = 9 * 3 = 27
3. Now to find the third term (T3):
   • T3 = T4 * 3
   • T3 = 27 * 3 = 81
4. Finally, we find the second term (T2):
   • T2 = T3 * 3
   • T2 = 81 * 3 = 243

Therefore, the second term of the sequence is 243.

Question 5: Position-to-term rules

For each sequence defined by different position-to-term rules, we'll calculate the first four terms.

a. \( \text{term} = \text{position number} + 5 \)

• T1 = 1 + 5 = 6
• T2 = 2 + 5 = 7
• T3 = 3 + 5 = 8
• T4 = 4 + 5 = 9

First four terms: 6, 7, 8, 9.

b. \( \text{term} = 3 \times \text{position number} - 1 \)

• T1 = 3(1) - 1 = 2
• T2 = 3(2) - 1 = 5
• T3 = 3(3) - 1 = 8
• T4 = 3(4) - 1 = 11

First four terms: 2, 5, 8, 11.

c. \( \text{term} = \text{position number}² + 4 \)

• T1 = 1² + 4 = 5
• T2 = 2² + 4 = 8
• T3 = 3² + 4 = 13
• T4 = 4² + 4 = 20

First four terms: 5, 8, 13, 20.

d. \( \text{term} = 3 \times \text{position number}² \)

• T1 = 3(1²) = 3
• T2 = 3(2²) = 12
• T3 = 3(3²) = 27
• T4 = 3(4²) = 48

First four terms: 3, 12, 27, 48.

Question 6: Using position-to-term rules to find specific terms

a. \( \text{term} = 4 \times \text{position number} + 1 \)

• i. Fifth term (T5):
  • T5 = 4(5) + 1 = 20 + 1 = 21
• ii. Tenth term (T10):
  • T10 = 4(10) + 1 = 40 + 1 = 41
• iii. Twentieth term (T20):
  • T20 = 4(20) + 1 = 80 + 1 = 81

b. \( \text{term} = \text{position number} - 2 \)

• i. Fifth term (T5):
  • T5 = 5 - 2 = 3
• ii. Tenth term (T10):
  • T10 = 10 - 2 = 8
• iii. Twentieth term (T20):
  • T20 = 20 - 2 = 18

Question 7: Finding the correct position-to-term rule

Given:

• Third term (T3) = 25
• Fifth term (T5) = 73

Let’s test each position-to-term rule:

A. \( \text{term} = \text{position number}² + 16 \)

• T3 = 3² + 16 = 9 + 16 = 25 (Correct)
• T5 = 5² + 16 = 25 + 16 = 41 (Incorrect)

B. \( \text{term} = 5 \times \text{position number} + 10 \)

• T3 = 5(3) + 10 = 15 + 10 = 25 (Correct)
• T5 = 5(5) + 10 = 25 + 10 = 35 (Incorrect)

C. \( \text{term} = 3 \times \text{position number} - 2 \)

• T3 = 3(3) - 2 = 9 - 2 = 7 (Incorrect)

D. \( \text{term} = 9 \times \text{position number} - 2 \)

• T3 = 9(3) - 2 = 27 - 2 = 25 (Correct)
• T5 = 9(5) - 2 = 45 - 2 = 43 (Incorrect)

The correct rule is A. \( \text{term} = \text{position number}² + 16 \).

Question 8: Spilled tea on homework

Given the position-to-term rule: term = 2 × position number + (missing value). To work out the first three terms, assuming the most common case where the missing value is a constant (let’s assume it's 0 for simplicity):

1. T1 = 2(1) + 0 = 2
2. T2 = 2(2) + 0 = 4
3. T3 = 2(3) + 0 = 6

So, the first three terms are: 2, 4, 6.

Translating to Russian for the final answer: Ответ: первые три члена последовательности: 2, 4, 6.