Question

The terms of a sequence are given recursively as \(a_{0}=0, a_{1}=4,\) and \(a_{n}=8 a_{n-1}-\) \(16 a_{n-2}\) for \(n \geq 2\). Find the first cight terms of this sequence.

Short answer

The first eight terms are 0, 4, 32, 192, 1024, 5120, 24576, and 114688.

Step-by-step solution

Understanding the initial terms

The sequence is defined recursively with initial terms given as \(a_0 = 0\) and \(a_1 = 4\). This means these values are already known and do not require further calculation.

Calculate the 3rd term (a_2)

To find \(a_2\), use the recursive formula \(a_n = 8a_{n-1} - 16a_{n-2}\). Substitute \(n=2\), which gives:\[a_2 = 8a_1 - 16a_0 = 8(4) - 16(0) = 32.\]

Calculate the 4th term (a_3)

Substitute \(n=3\) into the recursive formula:\[a_3 = 8a_2 - 16a_1 = 8(32) - 16(4) = 256 - 64 = 192.\]

Calculate the 5th term (a_4)

Substitute \(n=4\) into the recursive formula:\[a_4 = 8a_3 - 16a_2 = 8(192) - 16(32) = 1536 - 512 = 1024.\]

Calculate the 6th term (a_5)

Substitute \(n=5\) into the recursive formula:\[a_5 = 8a_4 - 16a_3 = 8(1024) - 16(192) = 8192 - 3072 = 5120.\]

Calculate the 7th term (a_6)

Substitute \(n=6\) into the recursive formula:\[a_6 = 8a_5 - 16a_4 = 8(5120) - 16(1024) = 40960 - 16384 = 24576.\]

Calculate the 8th term (a_7)

Substitute \(n=7\) into the recursive formula:\[a_7 = 8a_6 - 16a_5 = 8(24576) - 16(5120) = 196608 - 81920 = 114688.\]

Compile the first eight terms

List the terms calculated: \(a_0 = 0\), \(a_1 = 4\), \(a_2 = 32\), \(a_3 = 192\), \(a_4 = 1024\), \(a_5 = 5120\), \(a_6 = 24576\), \(a_7 = 114688\).

Key concepts

Initial Conditions

In the realm of recursive sequences, the first and most crucial step is to establish the initial conditions. This particular sequence starts with well-defined initial terms, specifically \(a_0 = 0\) and \(a_1 = 4\). These initial conditions serve as our starting point for generating the entire sequence.
Without these given terms, it would be impossible to calculate the subsequent values since each new term relies on the previous terms to compute the value. Think of initial conditions like the base of a building; every other layer depends on the integrity of this base.
They provide the necessary "fuel" to power the recurrence formula used to advance through the sequence.

• Initial conditions are provided values of the sequence.
• They are essential as a foundation for further calculations.
• In this exercise, \(a_0\) and \(a_1\) initiate the recursive process.

Recursive Formula

A recursive formula is the rule that defines each term of the sequence using one or more of its preceding terms. In the exercise, the formula is given as:
\[a_n = 8a_{n-1} - 16a_{n-2}\]
for \(n \geq 2\). This formula tells us how to find any term \(a_n\), as long as we know the previous two terms, \(a_{n-1}\) and \(a_{n-2}\).
The advantage of using a recursive formula is that it condenses what could be a long series of steps into a singular, powerful equation that generates an entire sequence with just initial conditions and a repeated calculation.

• It specifies how terms relate to each other, using prior terms in the computation.
• The recursive nature necessitates an understanding of the pattern from one term to the next.
• In our sequence, it requires the two previous terms to calculate the next term.

Term Calculation

To calculate terms in a recursively defined sequence, we utilize the recursive formula along with the initial conditions. Let's break down how this process works for our sequence.
Given the initial conditions, we start the calculation of subsequent terms from \(n = 2\). Each new term is found by substituting known values of earlier terms into the recursive formula.
For example, for \(a_2\), using \(a_1\) and \(a_0\), the result is \(8 \times 4 - 16 \times 0 = 32\). We then use already calculated terms to find \(a_3\), \(a_4\), and so forth.

• Each term builds on the calculated outcome of its predecessors.
• Accuracy in these calculations ensures that the sequence remains consistent.
• This process continues until all desired terms are calculated.

In essence, term calculation is both repetitive and straightforward once the recursive formula and initial conditions are in place.