Question

Write the fi rst six terms of the sequence. $$ \begin{aligned} &a_1=1 \\ &a_n=\left(a_{n-1}\right)^2-10 \end{aligned} $$

Short answer

The first six terms of the sequence are: 1, -9, 71, 5031, 25310611, 640164383788511

Step-by-step solution

Calculate the second term

Given the first term \(a_1 = 1\), we substitute \(n = 2\) into the formula \(a_{n} = (a_{n-1})^2 - 10\), this gives us: \(a_{2} = (a_{1})^2 - 10 = (1)^2 - 10 = 1 - 10 = -9\)

Calculate the third term

Now, we have the second term as \(a_2 = -9\). We substitute \(n = 3\) in our formula: \(a_{3} = (a_{2})^2 - 10 = (-9)^2 - 10 = 81 - 10 = 71\)

Calculate the fourth term

With the third term as \(a_3 = 71\), input \(n = 4\) into the formula: \(a_{4} = (a_{3})^2 - 10 = (71)^2 - 10 = 5041 - 10 = 5031\)

Calculate the fifth term

Then we have \(a_4 = 5031\) and substitute \(n = 5\) into our formula: \(a_{5} = (a_{4})^2 - 10 = (5031)^2 - 10 = 25310621 - 10 = 25310611\)

Calculate the sixth term

Finally, using the fifth term \(a_5 = 25310611\), substitute \(n = 6\) into the formula: \(a_{6} = (a_{5})^2 - 10 = (25310611)^2 - 10 = 640164383788521 - 10 = 640164383788511\)

Key concepts

Recursive Sequences

A recursive sequence is a sequence where each term depends on one or more of its preceding terms. It is like a chain reaction; each term keeps the sequence going by building on the previous one. Let's take our given example. The first term, or seed, is provided: \(a_1 = 1\). From this initial element, the whole sequence can be generated. For every subsequent term \(a_n\), the formula \(a_n = (a_{n-1})^2 - 10\) is used. Here, \(a_{n-1}\) represents the previous term, ensuring that each new term depends directly on the term before it, thus forming a chain.
This type of sequence helps in predicting future terms without listing the entire sequence. Thus, understanding the starting term and the recursive rule is key to finding any term in the sequence.

Mathematical Induction

Mathematical induction is a powerful method of proof in mathematics used to establish that a statement holds true for all natural numbers. It involves two main steps: the base case and the inductive step.
The base case checks if the statement is true for the initial value, typically \(n = 1\). For our sequence, we start with \(a_1 = 1\), and using our formula, we find \(a_2, a_3,\) and so on.
The inductive step assumes the statement holds for some arbitrary number \(k\) and proves it for \(k+1\). If it's shown to be true for \(k+1\), it must be true for all subsequent numbers. This technique is not directly used in the computation of sequences here, but it's essential in validating the process or constructing new sequences.

Sequence Formula

The sequence formula provides a mechanism to find any term in a sequence without listing all prior terms. For recursive sequences, the formula gives a rule based on previous terms. In this exercise, the formula \(a_n = (a_{n-1})^2 - 10\) provides a direct calculation from the prior term.
While this approach involves calculating each term step by step, a sequence formula enables these calculations to continue indefinitely, ensuring comprehension of sequence behavior. Recognizing and interpreting sequence formulas open doors to predicting patterns and behaviors in varying sequences.

• Initial term and calculations form the sequence.
• Computational procedures build on previous results.

Algebra 2

In Algebra 2, sequences are a foundational concept often revisited to build more complex ideas. This includes understanding sequences like arithmetic and geometric sequences, with recursive sequences as a key type.
Recursive sequences demand varied understanding facets, including manipulation and practical applications, which open a profusion of problem-solving techniques in math. This sequence example practically uses squaring in recursion, developing large numbers very quickly.
Algebra 2 presents students with diverse aspects, such as understanding foundational operations, solving sequences, and recognizing growth patterns. These principles form the basis of advanced algebraic thinking utilized in predicting outcomes and simplifying complex mathematics.