Problem 24 - Precalculus: Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is geometric. If it is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1}.\) $$a_{n}=4+3^{n}$$

Step-by-Step Solution

Answer

The sequence is not geometric.

Step 1: Calculate the First Five Terms

The given sequence is defined by the formula \(a_{n} = 4 + 3^{n}\). To find the first five terms, compute \(a_{1}, a_{2}, a_{3}, a_{4},\) and \(a_{5}\) by substituting \(n = 1, 2, 3, 4,\) and \(5\) into the formula. - For \(a_{1}\): \(a_{1} = 4 + 3^{1} = 4 + 3 = 7\).- For \(a_{2}\): \(a_{2} = 4 + 3^{2} = 4 + 9 = 13\).- For \(a_{3}\): \(a_{3} = 4 + 3^{3} = 4 + 27 = 31\).- For \(a_{4}\): \(a_{4} = 4 + 3^{4} = 4 + 81 = 85\).- For \(a_{5}\): \(a_{5} = 4 + 3^{5} = 4 + 243 = 247\). Therefore, the first five terms are 7, 13, 31, 85, and 247.

Step 2: Determine If the Sequence is Geometric

A sequence is geometric if the ratio of consecutive terms is constant. Calculate the ratios \(\frac{a_{2}}{a_{1}}\), \(\frac{a_{3}}{a_{2}}\), \(\frac{a_{4}}{a_{3}}\), and \(\frac{a_{5}}{a_{4}}\):- \(\frac{a_{2}}{a_{1}} = \frac{13}{7}\approx1.86\).- \(\frac{a_{3}}{a_{2}} = \frac{31}{13}\approx2.38\).- \(\frac{a_{4}}{a_{3}} = \frac{85}{31}\approx2.74\).- \(\frac{a_{5}}{a_{4}} = \frac{247}{85}\approx2.91\).Since the ratios are not constant, the sequence is not geometric.

Key Concepts

Geometric Sequences

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the *common ratio*. This common ratio remains the same for every two consecutive terms in the sequence. In math terms, you could express a geometric sequence as: - First term = \( a \) - Common ratio = \( r \)- nth term = \( a_n = a \cdot r^{n-1} \)This formula is powerful as it allows you to find any term in the sequence quickly without having to list all previous terms. Geometric sequences are common in various real-world applications, like calculating compound interest or modeling exponential growth sequences in populations. In the provided exercise, however, the sequence was not geometric because the ratio between consecutive terms was not constant. The differences in the calculated ratios such as \( \frac{13}{7} \) and \( \frac{31}{13} \) illustrate this inconsistency.

Sequence Terms

The term *sequence terms* refers to the individual numbers or expressions in a sequence. The terms of a sequence are often denoted as \( a_1, a_2, a_3, \ldots, a_n \), where each subscript number represents the position of the term within the sequence. Understanding sequence terms is crucial when working through problems like the one in the exercise.In this exercise, the sequence is expressed by the formula \( a_n = 4 + 3^n \), which specifies how to calculate each term in the sequence. By substituting the desired term number into the formula, you can calculate:- \( a_1 = 7 \)- \( a_2 = 13 \)- \( a_3 = 31 \)- \( a_4 = 85 \)- \( a_5 = 247 \)This step-by-step calculation of sequence terms helps in assessing whether a sequence follows any patterns, such as being arithmetic or geometric, and understanding the behavior of the sequence.

Common Ratio

The *common ratio* is specific to geometric sequences and is the fixed factor that each term is multiplied by to get the next term. Denoted by the letter \( r \), it can be calculated by dividing any term by its preceding term. Geometric sequences rely on this unchanging ratio to maintain their uniform pattern of increase or decrease.Using this concept in the context of the exercise, students calculated the ratios between successive terms. However, these did not stay constant:- \( \frac{13}{7} \approx 1.86 \)- \( \frac{31}{13} \approx 2.38 \)- \( \frac{85}{31} \approx 2.74 \)- \( \frac{247}{85} \approx 2.91 \)These varying numbers demonstrate that the sequence doesn't have a common ratio and therefore cannot be classified as a geometric sequence. Recognizing whether a sequence is geometric helps decide if the sequence can be expressed in the specific form \( a_n = a \cdot r^{n-1} \), using its first term and common ratio.