Problem 61 - Algebra for College Students

Question

$$ \sum_{=1}^{30}(-2 i+4)-810 $$

Step-by-Step Solution

Answer

The expression evaluates to -1620.

Step 1: Simplify the Summation Expression

The given expression is \( \sum_{i=1}^{30}(-2i + 4) \). This represents the sum of the sequence where each term is defined as \(-2i + 4\). First, identify the sequence's first few terms to understand the pattern:- When \( i = 1 \): Term = \(-2(1) + 4 = 2\)- When \( i = 2 \): Term = \(-2(2) + 4 = 0\)- When \( i = 3 \): Term = \(-2(3) + 4 = -2\)The sequence decreases by 2 each term.

Step 2: Calculate the Sum of the Sequence

Now, we can calculate the sum of this sequence. Notice it is an arithmetic sequence, with a first term \( a_1 = 2 \) and a common difference \( d = -2 \). The number of terms \( n = 30 \).The nth term (\( a_n \)) of an arithmetic sequence is defined as:\[ a_n = a_1 + (n-1) imes d \]Substituting given values:\[ a_{30} = 2 + (30-1) imes (-2) = 2 - 58 = -56 \]The sum \( S_n \) of an arithmetic sequence is given by:\[ S_n = \frac{n}{2} imes (a_1 + a_n) \]Substitute the known values:\[ S_{30} = \frac{30}{2} imes (2 + (-56)) = 15 imes (-54) = -810 \]

Step 3: Subtract 810 from the Summation

We now need to evaluate the expression \( \sum_{i=1}^{30}(-2i + 4) - 810 \). From the previous step, we found that the sum of the sequence is \(-810\).Thus, plug it into the expression:\(-810 - 810 = -1620\).

Key Concepts

Sum of Arithmetic Sequence

Arithmetic sequences are a fundamental topic in mathematics, where each term differs from the previous one by a constant value, known as the common difference. Understanding how to compute the sum of such sequences is useful in many practical situations.

For an arithmetic sequence, the sum of the sequence can be determined using a specific formula:

• Identify the number of terms, often denoted as \( n \).
• Find the first term, \( a_1 \), and the last term, \( a_n \).
• Use the sum formula: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]

This formula works by essentially averaging the first and last terms, and then multiplying it by the number of terms in the series.

In our given example, the sequence starts at 2 and ends at -56, with a total of 30 terms. Plugging into the formula, we calculated that the sum is \(-810\). This helps consolidate the understanding that arithmetic sequences follow quite predictable patterns, making calculations manageable.

Sequence Patterns

Grasping the pattern in a sequence is a stepping stone to solving arithmetic sequence problems. A sequence pattern reveals how the terms change, confirming its nature as either arithmetic or otherwise.

In an arithmetic sequence, each term is created by adding or subtracting a constant value (called common difference) to the previous term.

• Calculate the initial few terms to confirm the nature of the sequence.
• Establish a common difference \( d \) by subtracting successive terms: \( a_2 - a_1 \).

In the problem provided, when plugging different integers for \(i\), such as \( i = 1, 2, 3\), we confirm a consecutive decrease of 2 in each term: 2, 0, -2. Recognizing this arithmetic pattern helps in applying the appropriate formulas for further computations. By acknowledging this decrease pattern, solving related problems becomes effortless.

Summation Notation

Summation notation offers a succinct way to express the sum of several terms in a sequence, using the Greek letter Sigma \( \Sigma \). This notation is robust and versatile, commonly employed in arithmetic and geometric sequences.
The essential components of a summation expression include:

• The starting index, often written below the \( \Sigma \).
• The ending index, positioned above the \( \Sigma \).
• The expression to be summed, written to the right of \( \Sigma \).

For instance, \( \sum_{i=1}^{30}(-2i + 4) \) implies the sum of terms from \( i = 1 \) to \( i = 30 \) for the expression \( -2i + 4 \). Each substitution of \( i \) generates a term in the sequence, which are then systematically summed.
Using summation notation not only simplifies complex calculations but also clarifies the components and their role in the calculation, thus providing a clean and efficient representation of the mathematical process.