Problem 13 - Algebra and Trigonometry

Question

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=7 \text { and } a_{n}=a_{n-1}+5 \text { for } n \geq 2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 7, 12, 17, and 22.

Step 1: Find the First Term

The first term \(a_1\) is directly given as 7. So, \(a_1 = 7\).

Step 2: Find the Second Term

For the second term \(a_2\), use the formula \(a_{n}=a_{n-1}+5\). Substitute \(n=2\) and \(a_{1}=7\) into the formula to find \(a_2\). Hence \(a_2 = a_{2-1} + 5 = a_1 + 5 = 7 + 5 = 12\).

Step 3: Find the Third Term

For the third term \(a_3\), substitute \(n=3\) and \(a_{2}=12\) into the formula to find \(a_3\). Hence \(a_3 = a_{3-1} + 5 = a_2 + 5 = 12 + 5 = 17\).

Step 4: Find the Fourth Term

For the fourth term \(a_4\), substitute \(n=4\) and \(a_{3}=17\) into the formula to find \(a_4\). Hence \(a_4 = a_{4-1} + 5 = a_3 + 5 = 17 + 5 = 22\).

Key Concepts

Recursion Formulas

Recursion is a mathematical process where a sequence of numbers is generated using a specific rule or formula that refers back to previous terms. A recursive formula provides a method to calculate the nth term of a sequence using the values of one or more preceding terms. It generally includes an initial term (or initial conditions) and a rule for determining subsequent terms.

For instance, in the given problem, we have the recursive formula \(a_n = a_{n-1} + 5\) for \(n \geq 2\), with the initial condition \(a_1 = 7\). The first part, \(a_n = a_{n-1} + 5\), tells us how to find any term in the sequence after the first term—by adding 5 to the previous term. The second part, \(a_1 = 7\), provides us with a concrete starting point.

Using recursive formulas, the calculation is done step by step, where each subsequent term relies on the computation of the previous term, and so it forms a chain of dependencies throughout the sequence.

Arithmetic Sequences

Defining an Arithmetic Sequence

The sequence given in the exercise is a perfect example of an arithmetic sequence. Such sequences have a common difference between consecutive terms, which in this case, is the number 5. An arithmetic sequence can be recognized by adding (or subtracting) the same value each time to get the next term.

So, the sequence produced by the given formula would be: \(a_1 = 7\), \(a_2 = 12\), \(a_3 = 17\), \(a_4 = 22\), and so on. Each term increases by 5 from the previous one, indicating a constant rate of change, a hallmark of arithmetic sequences. This makes terms prediction and calculation manageable and explains why such sequences are commonly taught in algebra.

Sequence Terms Calculation

When it comes to calculating the terms of a sequence, the steps are often straightforward but require attention to detail. In the case of the exercise, the first term of the sequence is given, and we use the recursion formula to calculate each subsequent term. It's essential to carefully substitute the correct values for each step.

To illustrate further sequence term calculation with an example:

• First Term \(a_1\): Directly known, in our case, \(a_1 = 7\).
• Second Term \(a_2\): Apply recursion formula with \(n=2\), resulting in \(a_2 = a_1 + 5 = 12\).
• Continuing in this manner, the third and fourth terms (\(a_3 = 17\) and \(a_4 = 22\)) are found by applying the same formula with \(n=3\) and \(n=4\), respectively.

Remember, precision in the substitution process is crucial; a mix-up in the terms can lead to incorrect results. By understanding and applying the formula methodically, the sequence terms can be calculated with ease.