Problem 17 - Algebra and Trigonometry

Question

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=4 \text { and } a_{n}=2 a_{n-1}+3 \text { for } n \geq 2 $$

Step-by-Step Solution

Answer

The first four terms of the given recursive sequence are 4, 11, 25 and 53.

Step 1: Compute the first term

The first term of the sequence, \( a_{1} \), is given as 4. So no calculation is required for this term, \( a_{1}=4 \).

Step 2: Compute the second term

To find the second term, \( a_{2} \), we need to substitute \( n=2 \) into the formula \( a_{n}=2a_{n-1}+3 \). As a result we obtain \( a_{2}=2a_{1}+3=2*4+3=11 \). So the second term, \( a_{2} \), of our sequence is 11.

Step 3: Compute the third term

The third term, \( a_{3} \), can be computed similarly to the second term. We just need to substitute \( n=3 \) into the formula to give us \( a_{3}=2a_{2}+3=2*11+3=25 \). So the third term, \( a_{3} \), of the sequence is 25.

Step 4: Compute the fourth term

Again, substituting \( n=4 \) into the formula gives us \( a_{4}=2a_{3}+3=2*25+3=53 \). So \( a_{4} \), the fourth term of the sequence, is 53.

Key Concepts

Sequence Terms

In mathematics, a sequence is a list of numbers in a specific order. Each individual number in a sequence is called a 'sequence term'.
Sequence terms follow a particular pattern or rule, which dictates the sequence's overall behavior. In our example, the sequence starts with a specific given term, known as the first term, which is \( a_1 = 4 \). This serves as the starting point from which subsequent terms are generated.
Understanding sequence terms is crucial because these terms embody the concrete values produced from the sequence. They are the numerical outputs from applying the defining rule of the sequence. In our problem, the first four terms are \( 4, 11, 25, \text{ and } 53 \). Each term depends on the previous term, constructing a chain of numbers that grow based on the pattern established by the recursion formula.

Recursion Formula

A recursion formula is a specific type of rule that defines each term of a sequence using previous terms. This means to find a new term, you use the term or terms before it.
In the given exercise, the recursion formula is \( a_{n} = 2a_{n-1} + 3 \) for \( n \geq 2 \). This formula tells us exactly how to calculate each new term from the one immediately before it, starting with the initial term that is often given or calculated separately.

• "\( 2a_{n-1} + 3 \)" implies that each term is two times the previous term plus 3.
• The formula applies to all terms beyond the first because it requires the previous term.

Recursion formulas are particularly powerful for defining sequences where the relationship between terms can be neatly expressed through earlier terms, allowing the sequence to grow in a consistent pattern.

Arithmetic Operations

Arithmetic operations are basic computations involving numbers, such as addition, subtraction, multiplication, and division.
These operations are used in recursive sequences to determine the next term from the previous one. In our recursion formula \( a_{n} = 2a_{n-1} + 3 \), each new term is derived by performing arithmetic operations on the previous term.

• The term \( 2a_{n-1} \) involves multiplying the previous term \( a_{n-1} \) by 2, an example of multiplication.
• Adding 3 is another arithmetic operation applied to the multiplied result.

These simple arithmetic steps help maintain the integrity of the sequence's growth pattern, enabling a clear and systematic way to predict and compute subsequent terms. By understanding how arithmetic operations apply within recursion formulas, it's easier to grasp how each term contributes to the sequence as a whole.