Problem 18 - Basic Engineering Mathematics

Question

Determine the tenth term of the series 3,6, \(12,24, \ldots\)

Step-by-Step Solution

Answer

The tenth term is 1536.

Step 1: Identify the Pattern

First, we need to understand how the series progresses. Looking at the first few terms, 3, 6, 12, 24, we observe that each term is double the previous term.

Step 2: Define the General Formula

Since each term is double the previous term, we define the nth term of the series with the formula: \( a_n = 3 \times 2^{(n-1)} \). This is because the first term is 3 (\(a_1 = 3 \times 2^0 = 3\)), the second term is 6 (\(a_2 = 3 \times 2^1 = 6\)), and so on.

Step 3: Substitute to Find the Tenth Term

We use the formula for the nth term to find the tenth term: \( a_{10} = 3 \times 2^{(10-1)} = 3 \times 2^9 \).

Step 4: Calculate the Tenth Term

Multiply the numbers to find the tenth term: \( 2^9 = 512 \). Then, multiply by 3 to find: \( a_{10} = 3 \times 512 = 1536 \).

Key Concepts

Sequence

In mathematics, a sequence is a list of numbers arranged in a specific order, following a particular rule or pattern. Each number in a sequence is called a term. For the series 3, 6, 12, 24, you may notice a consistent pattern in the increase from one term to the next.

To understand sequences, consider each term as a step guided by a particular rule. In this exercise, the rule is quite simple: each term is double the previous term.

• The first term is 3.
• Then, double it to get 6.
• Double 6 to get 12, and so on.

This predictable doubling pattern is what defines this series as a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a fixed number, known as the common ratio. In this case, the common ratio is 2.

Exponential Function

Exponential functions are mathematical expressions involving powers, or exponents, of a constant base. They describe growth or decay where the rate of change is proportional to the current value.

In the given series, the exponential function comes into play in defining the general nth term of the sequence. It uses the idea that each term grows exponentially based on its position in the sequence. This is where powers of two (2 raised to some exponent) become crucial.

• The general formula is given as \( a_n = 3 \times 2^{(n-1)} \).
• Here, 3 is the initial term or the base and \(2^{(n-1)}\) is the exponential part.

Notice that the exponent \(n-1\) reflects the term's position minus one—starting from zero for the first term. This reflects how each term increases exponentially based on its ordinal number.

Mathematical Formula

A mathematical formula is a concise way of expressing information symbolically, often in the form of an equation. In the context of a geometric series, the formula is essential for predicting terms based on their position within the sequence.

For this particular exercise, the formula \( a_n = 3 \times 2^{(n-1)} \) allows us to determine any term in the sequence without having to calculate all preceding terms. Here's how it is integrated and works:

• \(a_1 = 3 \times 2^0 = 3\) for the first term.
• \(a_2 = 3 \times 2^1 = 6\) for the second term, confirming that the sequence doubles.
• To find the tenth term \(a_{10}\), plug in \( n = 10\), yielding \(a_{10} = 3 \times 2^9 = 1536\).

This formula exploits the constant ratio to quickly and accurately find any term, emphasizing the power of mathematical expressions to simplify complex calculations.