Problem 43 - 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. $$ 1^{2}+2^{2}+3^{2}+\cdots+15^{2} $$

Step-by-Step Solution

Answer

The series can be written in summation notation as \( \sum_{i =1}^{15} i^2 \).

Step 1: Identify the pattern/series in the expression

From the expression, we observe that each term is a square of an integer, starting from 1 to 15. This is a series of the form \(n^2\).

Step 2: Express the series in summation notation

The sigma notation (or summation notation) for a series is expressed as \( \sum_{i =a}^{b} f(i) \), where \( 'a' \) is the starting value, \( 'b' \) is the end value, and \( 'f(i)' \) is the function of \( 'i' \) defining the term in the series. Here, our 'a' is 1, 'b' is 15, and 'f(i)' is \( i^2 \). Thus the series can be written in sigma notation as \( \sum_{i =1}^{15} i^2 \).

Key Concepts

Series

Often, we encounter a list of numbers added together. This is what we call a series. In this case, our list includes the squares of whole numbers. A series can be thought of as a sum of elements that follow a specific pattern. For instance, given a list of squares starting at 1 and going up to 15, the series becomes:

• 1², 2², 3², up to 15².

This is simply a concise way to write the sum of all these squared numbers. Series help simplify expressions, making it easier to tell what needs to be summed without writing it out in full.
Understanding series can help solve mathematical problems that require summation, pattern recognition, or analysis of numbers in sequence.

Sigma Notation

Sigma notation is a mathematical symbol used to express a series. It streamlines lengthy and tedious number patterns into an easy-to-read format. The symbol 'Σ' (sigma) means to sum up over a range of numbers.
Here's how it breaks down:

• The expression below it defines where the summation starts, known as the lower limit.
• The expression above it tells us where the summation ends, known as the upper limit.
• The expression next to the sigma shows what terms are being added.

For our series 1² + 2² + ... + 15², sigma notation is given by \[ \sum_{i=1}^{15} i^2 \].This tells us to sum squared numbers starting from i = 1 up to i = 15. Sigma notation allows mathematicians to express large sums succinctly and efficiently.

Sequence

A sequence is a set of numbers in a specific order. Unlike a series, which sums the numbers, a sequence simply lists them. The sequence in our exercise is the squared numbers: 1², 2², 3², ..., 15².
Some characteristics include:

• Order: Sequences need a specific pattern or rule, like the squares of integers.
• Terms: Each number in the sequence, such as 1² or 4², is a term.

Recognizing sequences is important for identifying relationships among numbers, which can help when tackling problems in calculus and algebra. Sequences often transition into series when summed.

Index of Summation

The index of summation is a placeholder variable used in sigma notation to signify which term is being considered. It is typically represented by 'i', 'j', or 'k'. In our case, 'i' is used as the index.

• The index identifies each term in a sequence or series step by step.
• It starts at a designated starting point (lower limit) and ends at a specified endpoint (upper limit).

For \[ \sum_{i=1}^{15} i^2 \], 'i' helps refer to elements from 1 through 15 within the context of the squared numbers. Adjusting the index of summation can redefine the scope of the series you are working with. It is crucial for accurately summing the terms in a series.