Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{2^{n+1}}{2^{n}+1}\\\ &a_{7}= \end{aligned}$$

Step-by-Step Solution

Answer

The result of \(a_{7}\) is \(\frac{256}{129}\)

Step 1: Substitute the value for n

Firstly, the value of \(n = 7\) should be substituted into the sequence function to get: \[a_{7}=\frac{2^{7+1}}{2^{7}+1}\]

Step 2: Simplify the exponent

Simplify the exponents in the expression to get: \[a_{7}=\frac{2^{8}}{2^{7}+1}\]

Step 3: Evaluate the exponent

Evaluate \(2^{8}\) and \(2^{7}\) to get: \[a_{7}=\frac{256}{128+1}\]

Step 4: Finalize the calculation

Simplify the denominator to get \[a_{7}=\frac{256}{129}\]

Key Concepts

Understanding Geometric Sequences

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

For example, in the sequence 2, 4, 8, 16, ..., each term is 2 times the previous term, making it a geometric sequence with a common ratio of 2. Geometric sequences can be decreasing as well, such as 100, 10, 1, 0.1, ..., where the ratio is 0.1. To express a general term of a geometric sequence, we use the formula: \[a_n = a_1 \times r^{(n-1)}\]where \(a_n\) is the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.

Understanding geometric sequences is crucial for solving many types of problems, including those involving exponential growth or decay, like population models or radioactive decay.

Exponent Simplification

Exponent simplification is an important concept when dealing with sequences, as it allows us to calculate terms more easily by using exponent rules.

One such rule is the power of a power rule, which states that when you raise a power to another power, you multiply the exponents: \[(a^m)^n = a^{m \times n}\].

Another important rule is the quotient of powers rule, which says that when you divide powers with the same base, you subtract the exponents: \[\frac{a^m}{a^n} = a^{m - n}\].

Simplifying exponents makes complex calculations more manageable and is essential for working through algebraic expressions, sequences, and scientific formulas.

Sequence Term Calculation

Sequence term calculation involves finding a specific term in a sequence, usually denoted as \(a_n\), where \(n\) is the position in the sequence.

To calculate the nth term of a geometric sequence, we can either use the explicit formula mentioned earlier or sequentially multiply the common ratio from one term to the next until we reach the nth term.

The ability to calculate sequence terms is especially useful in predictive models where you might want to know the value of a certain term far into the sequence without having to list out all preceding terms. This is not only a time saver but also a way to apply sequences to real-world scenarios, like computing future investment values or predicting patterns in data.