The image shows a problem written on a blackboard. The text appears to describe the conditions of a sequence or pattern. Here's the transcription of the problem:

1. "The first term is 2 and the pattern is multiplied by 5, minus 7."
2. "Starts with 10, and the pattern is 'half of the current number plus 8'."

Let's analyze and break down the problem:

Analysis:

• First Sequence:

  • The first term is given as 2.
  • The pattern involves multiplying the current term by 5 and then subtracting 7.

  So, if $a_n$ is the $n$-th term of the sequence: $a_{n+1} = 5 \times a_n - 7$ The first few terms can be calculated as:

  • $a_1 = 2$
  • $a_2 = 5 \times 2 - 7 = 10 - 7 = 3$
  • $a_3 = 5 \times 3 - 7 = 15 - 7 = 8$
  • $a_4 = 5 \times 8 - 7 = 40 - 7 = 33$
  • and so on.

• Second Sequence:

  • Starts with 10.
  • The pattern is described as "half of the current number plus 8."

  If $b_n$ is the $n$-th term of the second sequence: $b_{n+1} = \frac{1}{2} b_n + 8$ The first few terms can be calculated as:

  • $b_1 = 10$
  • $b_2 = \frac{1}{2} \times 10 + 8 = 5 + 8 = 13$
  • $b_3 = \frac{1}{2} \times 13 + 8 = 6.5 + 8 = 14.5$
  • $b_4 = \frac{1}{2} \times 14.5 + 8 = 7.25 + 8 = 15.25$
  • and so on.

Would you like further details or have any questions?

Related Questions:

1. What is the general formula for the sequence given the recursive relation?
2. How would you find the 10th term of the first sequence?
3. Can you determine if the first sequence is arithmetic or geometric?
4. How would you find the sum of the first 5 terms of the second sequence?
5. What happens to the second sequence as the number of terms increases?
6. Is there a fixed point for the second sequence where the sequence stabilizes?
7. How can these sequences be represented graphically?
8. What is the importance of understanding recursive sequences in mathematics?

Tip:

When dealing with sequences, especially recursive ones, it's useful to calculate a few initial terms to identify patterns or behaviors, such as convergence or divergence.