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A recursive formula is a way to define a sequence where each term is formulated based on the previous ones. This method is crucial in many mathematical sequences as it provides a straightforward way to predict future terms without relying just on a general formula.

In this exercise, we have two sequences: \( a_n \) and \( b_n \), where each follows its specific recursive formula. For \( a_n \), the recursive formula is given as \( a_n = a_{n-1} + 3n \). This means to find any term \( a_n \), you simply take the previous term \( a_{n-1} \) and add \( 3n \) to it.

For sequence \( b_n \), the recursive relationship is \( b_n = b_{n-1} + a_{n-1} \). This shows that each term in sequence \( b_n \) is formed by adding the preceding term \( b_{n-1} \) and the \( a_{n-1} \) term from sequence \( a_n \). This interconnectedness in the sequences showcases the power of recursive formulas.

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. This constant is called the "common difference."

In our exercise, the sequence \( a_n \) is not strictly an arithmetic sequence due to the variable factor \( 3n \). Though each term relies on the previous, the additional multiplier \( n \) makes \( a_n \) grow faster than a typical arithmetic sequence. However, as a part of generating each term, it does follow a rule that can remind one of arithmetic sequences due to the addition aspect of the formula.

Sequence \( b_n \) is linked to \( a_n \), capturing its essence and expanding it by adding previous terms from both sequences. This creates an interesting dynamic that challenges our typical understanding of arithmetic sequences.

To solve the initial exercise, we need to find the values of \( b_2, b_3, b_4, \) and \( b_5 \). The calculation relies on understanding the recursive approach to each sequence term.

**Calculation Steps:**

• Finding \( a_2 \): Start with \( a_1 = 8 \), use \( a_2 = a_1 + 3 \times 2 = 14 \).
• Finding \( b_2 \): With \( b_1 = 5 \) and \( a_1 = 8 \), find \( b_2 = b_1 + a_1 = 13 \).

Continue this pattern:

• Compute \( a_3 \): \( a_3 = a_2 + 3 \times 3 = 23 \).
• Calculate \( b_3 \): \( b_3 = b_2 + a_2 = 27 \).
• Find \( a_4 \): \( a_4 = a_3 + 3 \times 4 = 35 \).
• Decipher \( b_4 \): \( b_4 = b_3 + a_3 = 50 \).
• Determine \( a_5 \): \( a_5 = a_4 + 3 \times 5 = 50 \).
• Uncover \( b_5 \): \( b_5 = b_4 + a_4 = 85 \).

This structured approach, using recursive calculations for both sequences, allows for meticulous and clear progression through the exercise.