Question

Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{1}=1, a_{2}=1 \text { and } a_{n+2}=a_{n}+a_{n+1} \text { for } n \geq 1 $$

Short answer

The first six terms of the sequence are 1, 1, 2, 3, 5, and 8.

Step-by-step solution

Understanding the Problem

We are given the recurrence relation for the sequence: \(a_1 = 1\), \(a_2 = 1\) and \(a_{n+2} = a_n + a_{n+1}\) for \(n \geq 1\). The task is to find the first six terms of this sequence starting from \(n = 1\).

Identify the First Two Terms

The first two terms are given directly as \(a_1 = 1\) and \(a_2 = 1\).

Calculate the Third Term

Using the recurrence relation \(a_{n+2} = a_n + a_{n+1}\), find the third term: \(a_3 = a_1 + a_2 = 1 + 1 = 2\).

Calculate the Fourth Term

Continue using the recurrence relation to find the fourth term: \(a_4 = a_2 + a_3 = 1 + 2 = 3\).

Calculate the Fifth Term

Use the recurrence relation again to find the fifth term: \(a_5 = a_3 + a_4 = 2 + 3 = 5\).

Calculate the Sixth Term

Finally, apply the recurrence relation to find the sixth term: \(a_6 = a_4 + a_5 = 3 + 5 = 8\).

Key concepts

Recurrence Relation

A recurrence relation is a way of defining a sequence of numbers using prior terms in the sequence. In mathematics, it gives us a rule or a formula to generate a sequence by repeatedly applying this rule to get subsequent terms.
Recall the Fibonacci sequence where each term after the first two is the sum of the two preceding terms. This forms a classic example of a recurrence relation: each new term depends on the two terms before it. It's like a recipe, where you need certain ingredients (previous terms) to create your next dish (term).
The relation provided here is:

• Given:
  • \(a_1 = 1\)
  • \(a_2 = 1\)
• To find: \(a_{n+2} = a_n + a_{n+1}, n \geq 1\)

This tells us once we know a few terms, we can calculate as many as we want. Each step relies on known previous values, akin to building a tower by stacking blocks one after the other. The understanding of recurrence relations is vital as it's used extensively in computer science for algorithms and complex calculations.

Sequence Terms

A sequence is an ordered list of numbers, where each number is called a term. In our problem, sequence terms are generated using the recurrence relation given. Each term is derived mathematically from the preceding ones.
To help visualize:

• The sequence starts with two known terms: \(a_1 = 1\) and \(a_2 = 1\).
• Subsequent terms are calculated using the relation \(a_{n+2} = a_n + a_{n+1}\).
• This provides the first six terms: 1, 1, 2, 3, 5, and 8.

Thus, a series of numbers reveal a pattern which in this case corresponds to a famous sequence—Fibonacci. Understanding each term’s relation to its predecessors not only helps in predicting other terms but also enriches our knowledge of mathematical sequences.

Mathematical Induction

Mathematical induction is a proof technique often used to establish the truth of an infinite sequence of statements or formulae. Though the problem did not require proving the formulas by induction, understanding this method can greatly enhance comprehension in sequence-related problems.
Induction works in two core steps:

• Base Case: Verify the statement is true for the initial term (e.g., \(n = 1\)).
• Inductive Step: Assume the statement holds for some arbitrary term \(k\), and then prove it for \(k+1\).

If both the base case and the inductive step are true, we can conclude the statement is true for all natural numbers. This method is akin to climbing a ladder by taking one step at a time, ensuring each step is secure before moving to the next. Though not directly applied in finding sequence terms here, knowing induction is helpful for proving the correctness of recurrence sequences and verifying patterns.