Problem 44 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1^{4}+2^{4}+3^{4}+\dots+12^{4} $$

Step-by-Step Solution

Answer

The simplified form of the sum \(1^{4}+2^{4}+3^{4}+ \cdots +12^{4}\) using summation notation is \(Σ_{i=1}^{12} i^{4}\).

Step 1: Identifying the pattern in the sequence

Here, we have a series where each term is the fourth power of its respective position in the sequence. For example, the first term is \(1^{4}\), second term is \(2^{4}\), third term is \(3^{4}\) and so on till the twelfth term, which is \(12^{4}\).

Step 2: Identifying the limits for the series

The starting limit of the series is \(\1\), which is the number raised to the fourth power in the first term. The ending limit of the series is \(12\), which is the number raised to the fourth power in the last term.

Step 3: Expressing the sequence using summation notation

The general term for this sequence is the fourth power of the index (i). The summation goes from \(1\) to \(12\). This can be expressed using sigma notation as \(Σ_{i=1}^{12} i^{4}\) which means that you start with \(i = 1\) and increment \(i\) by one each time up to \(i = 12\), at each step adding \(i^{4}\) to the total.

Key Concepts

Sigma Notation

Sigma notation, often represented by the Greek letter Sigma \(\Sigma\), is a convenient and compact way of representing long sums. It allows mathematicians and students alike to write the addition of a sequence of numbers succinctly. The structure includes the sigma symbol \(\Sigma\), an expression for the terms to be added, and two numbers that define the bounds of the summation: the lower limit and the upper limit.

In the case of our exercise \(1^4 + 2^4 + 3^4 + \dots + 12^4\), the sigma notation simplifies this addition into one expression: \(\Sigma_{i=1}^{12} i^4\). The lower limit, 1, tells you where to start, and the upper limit, 12, tells you where to stop. The \(i\) underneath the \(\Sigma\) is called the index of summation, and \(^4\) indicates the operation to perform on each \(i\).

Mathematical Sequences

Mathematical sequences are ordered lists of numbers following a specific pattern or rule. These sequences show up repeatedly in various facets of mathematics. One important aspect of sequences is their ability to help us predict subsequent numbers in a series and to define them collectively.

In our example, the sequence is created by raising integers to the power of four: \(1^4\), \(2^4\), \(3^4\), and so on. Recognizing the pattern is pivotal in writing the series in summation notation, as it allows us to define the general term. For the given sequence, the pattern is that each term is the exponentiation of its position in the sequence to the fourth power.

Exponentiation

Exponentiation is the mathematical operation, written as \(b^n\), involving two numbers, the base \(b\) and the exponent or power \(n\). It signifies the operation of multiplying the base by itself, \(n\) times. In the language of our example, starting with the base 1 and incrementing it by 1 up to 12, each time the base is raised to the fourth power.In the series provided, the exponentiation serves to show progression and growth in the sequence, where the exponent determines how quickly the terms escalate. Terms like \(1^4\), \(2^4\), and \(3^4\) are computed as 1, 16, and 81, respectively, illustrating a rapid increase.

Series and Summation

Series and summation terms are often used interchangeably in mathematics, although a series typically refers to the sum of terms of a sequence. A series is the expression of the addition of the terms of a sequence, while summation is the act of calculating the total.

In the context of the given problem, we're dealing with a series – the sum of the fourth powers of the first twelve positive integers. The power of the summation notation lies in its ability to express the series as a whole, which can then be evaluated using various mathematical techniques or formulas specific to the type of series.Understanding the concept of series and summation, we can see how the series \(1^4 + 2^4 + 3^4 + \dots + 12^4\) serves as a finite sum of a sequence of terms defined by a clear rule, each term being a number exponentiated to the fourth.