Question

The terms of a sequence are given recursively as \(p_{0}=3, p_{1}=7,\) and \(p_{n}=3 p_{n-1}-\) \(2 p_{n-2}\) for \(n \geq 2\). Find the first eight terms of this sequence.

Short answer

The first eight terms of the sequence are 3, 7, 15, 31, 63, 127, 255, and 511.

Step-by-step solution

Identify the Initial Terms

Begin by identifying the given initial terms of the sequence. We have \( p_0 = 3 \) and \( p_1 = 7 \). These will be used as the starting point for finding subsequent terms.

Recursive Definition

Understand the recursive formula for generating the sequence: \( p_n = 3p_{n-1} - 2p_{n-2} \) for \( n \geq 2 \). This formula allows us to find each term based on the two preceding terms.

Calculate the Second Term

Since \( p_1 = 7 \) is given as an initial term, we start using the recursive formula to find the next terms from \( p_2 \). Applying the recursive formula:\[p_2 = 3p_1 - 2p_0 = 3(7) - 2(3) = 21 - 6 = 15.\]Hence, \( p_2 = 15 \).

Calculate the Third Term

Use \( p_2 \) and \( p_1 \) to find \( p_3 \) using the recursive formula:\[p_3 = 3p_2 - 2p_1 = 3(15) - 2(7) = 45 - 14 = 31.\]Thus, \( p_3 = 31 \).

Calculate the Fourth Term

Continue the pattern with \( p_3 \) and \( p_2 \):\[p_4 = 3p_3 - 2p_2 = 3(31) - 2(15) = 93 - 30 = 63.\]Therefore, \( p_4 = 63 \).

Calculate the Fifth Term

Next, apply the recursive relation once more using \( p_4 \) and \( p_3 \):\[p_5 = 3p_4 - 2p_3 = 3(63) - 2(31) = 189 - 62 = 127.\]So, \( p_5 = 127 \).

Calculate the Sixth Term

Utilizing the last two terms, \( p_5 \) and \( p_4 \):\[p_6 = 3p_5 - 2p_4 = 3(127) - 2(63) = 381 - 126 = 255.\]Thus, \( p_6 = 255 \).

Calculate the Seventh Term

Use \( p_6 \) and \( p_5 \) in the formula:\[p_7 = 3p_6 - 2p_5 = 3(255) - 2(127) = 765 - 254 = 511.\]Therefore, \( p_7 = 511 \).

Key concepts

Initial Terms in Sequences

In any sequence, identifying the initial terms is a critical first step. Think of these initial terms as the foundation of a building. If you're given a sequence in a problem as we are here, they are usually specified directly. For our sequence, we have two initial terms given:

• \( p_0 = 3 \)
• \( p_1 = 7 \)

These initial terms are like pieces of a puzzle or starting coordinates on a map, from which all other numbers in the sequence can be determined using a recursive formula. Remember, these initial terms are crucial because they provide the necessary "starting" information to use the recursive formula effectively. Without them, calculating the sequence accurately would be impossible.

Recursive Formula

A recursive formula in a sequence can be a fascinating concept. It is a rule that generates each term from one or more of its predecessors. In our sequence, the recursive formula is given by: \[p_n = 3p_{n-1} - 2p_{n-2}\]This equation shows that each term (\( p_n \)) is determined based on the two preceding terms (\( p_{n-1} \) and \( p_{n-2} \)).Here’s how the recursive formula works:

• "3" is a coefficient that multiplies the immediate previous term, \( p_{n-1} \).
• "2" is a coefficient that multiplies the term before the last, \( p_{n-2} \).
• The minus sign indicates a subtraction between these two products.

This type of formula is powerful because it allows you to build a sequence without having to express every term individually. It makes sequences dynamic, relying on their own structure to unfold.

Sequence Calculation Steps

Calculating terms in a recursive sequence involves a systematic approach, often breaking down into simple arithmetic based on the recursive formula.Here's how you calculate the terms:

• **Start with Initial Terms:** Remember that to even begin, you need the initial terms \( p_0 = 3 \) and \( p_1 = 7 \).
• **Apply the Recursive Formula:** Use it to find the next term, say \( p_2 \). This involves substituting known values:\[p_2 = 3 \cdot p_1 - 2 \cdot p_0 = 3 \cdot 7 - 2 \cdot 3 = 21 - 6 = 15\]
• **Proceed Sequentially:** Repeat the process. Each new term depends on the two terms directly before it. For instance:\[p_3 = 3 \cdot p_2 - 2 \cdot p_1 = 3 \cdot 15 - 2 \cdot 7 = 45 - 14 = 31\]Continue this method for each successive term until the desired number of terms (like the first eight in this case) is calculated.

The beauty of this process is in its iterative nature — you keep using known results to find new ones efficiently. Each step is like a link in a chain, strong only when connected to the last.