Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. Use the pattern to write the \(n\) th term of the sequence as a function of \(n .\) (Assume \(n\) begins with 1.) $$a_{1}=14, a_{k+1}=-2 a_{k}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are 14, -28, 56, -112 and 224. The formula for the n-th term of the sequence (where n starts at 1) is \(a_n = 14 \times (-2)^{n-1}\).

Step 1: Calculating the first five terms

Using the initial term \(a_{1}=14\) and the relation \(a_{k+1}=-2a_k\), we can calculate the subsequent terms one at a time:\n\nFor \(k=1\), substitute into the equation: \(a_{1+1}=-2a_1 = -2(14)=-28\). This gives \(a_2 = -28\).\n\nFor \(k=2\), again substitute: \(a_{2+1}= -2a_2=-2(-28) =56\). This gives \(a_3 = 56\).\n\nWe can continue this process to find \(a_4\) and \(a_5\), which are respectively -112 and 224.

Step 2: Deriving a formula for nth term

Looking at the calculated terms, we notice that each term is -2 times the preceding one. In other words, we are repeatedly multiplying initial term by \(-2\). This pattern resembles a geometric sequence with the first term \(a_{1}=14\) and a common ratio of \(-2\). The formula for the n-th term (where index starts from 1) of a geometric sequence is given as \(a = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. In this case, applying the formula, we have \(a_n = 14 \times (-2)^{(n - 1)}\).

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term after the initial is found by multiplying the previous term by a constant called the common ratio. The recursive pattern of multiplying by a fixed number creates a progression that can be visualized as steps that increase or decrease proportionally. In the given exercise, the sequence is generated by starting with 14 and repeatedly multiplying by -2, creating a sequence where each term changes sign and doubles in magnitude from the previous term.

To reinforce the understanding of this concept, let's visualize the sequence starting with 14: from there, each term is the product of the previous term and -2:

• 14 (Initial term, also called the first term)
• -28 (Second term found by multiplying the first term by the common ratio -2)
• 56 (Third term found by multiplying the second term by the common ratio -2)
• -112 (And so on...)
• 224 (Each term follows the pattern).

The ability to recognize this pattern is essential for understanding geometric sequences and for predicting future terms in the sequence.

Recursive Formula

A recursive formula is a formula that defines each term of a sequence using the preceding terms. Such formulas are invaluable for displaying the relationship between consecutive terms and for computer algorithms that generate sequences. In our specific exercise, the recursive relation is expressed as \(a_{k+1}=-2 a_{k}\), which concisely encapsulates how each term in the sequence is generated from the term before it. The initial term, also known as the seed of the sequence, needs to be known, which is provided as \(a_{1}=14\).

Understanding recursive formulas require students to consider each step sequentially rather than being able to jump to any term directly. This concept is fundamental when sequences are too complex for direct formulas or when learning programming, as it mirrors the iterative logic used in code loops.

Sequence Nth Term

Determining the \(n\)th term of a sequence is a common task in mathematics, as it allows evaluating the sequence at any given point without enumerating all previous terms. For geometric sequences, the \(n\)th term can be directly calculated using the formula \(a = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Applying this formula to our exercise, the \(n\)th term for the sequence is \(a_n = 14 \times (-2)^{(n - 1)}\). This compact expression allows students to bypass the recursive process and directly find the value of any term in the sequence, an efficiency that becomes particularly useful for large values of \(n\).

The ability to find the \(n\)th term is crucial in various mathematical and real-world applications, such as predicting population growth, calculating interest, or understanding patterns in natural phenomena.