Problem 66 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{6}\left(2+i-i^{2}\right)$$

Step-by-Step Solution

Answer

The sum of the series is -58.

Step 1: Write Out the Series

First, use the given formula to write out the terms of the series. Substitute each integer from 1 to 6 into the expression \(2 + i - i^2\).

Step 2: Calculate Each Term

Calculate each term in the series by substituting the values of \(i\) from 1 to 6 into the expression \(2 + i - i^2\). - For \(i = 1\): \(2 + 1 - 1^2 = 2 + 1 - 1 = 2\)- For \(i = 2\): \(2 + 2 - 2^2 = 2 + 2 - 4 = 0\)- For \(i = 3\): \(2 + 3 - 3^2 = 2 + 3 - 9 = -4\)- For \(i = 4\): \(2 + 4 - 4^2 = 2 + 4 - 16 = -10\)- For \(i = 5\): \(2 + 5 - 5^2 = 2 + 5 - 25 = -18\)- For \(i = 6\): \(2 + 6 - 6^2 = 2 + 6 - 36 = -28\)

Step 3: Add Up the Calculated Terms

Sum the calculated terms from Step 2 to get the total sum of the series:\[ 2 + 0 + (-4) + (-10) + (-18) + (-28) = -58 \]

Step 4: Verify the Calculation

Double-check the calculations by revisiting each term and ensuring the arithmetic is correct to confirm the series sum is \(-58\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the backbone of many mathematical equations and problems. In this exercise, the expression we are dealing with is \(2 + i - i^2\), which consists of three parts:

• The constant \(2\), which doesn't change.
• The linear term \(i\), which represents variable inputs from a specific set of numbers.
• The quadratic term \(-i^2\), which introduces non-linearity, as it squares the variable and multiplies it by \(-1\).

Algebraic expressions can be evaluated by substituting the values of variables (like \(i\)) with numbers. When substituting, it’s critical to respect operation order: first, handle exponents, then multiplication/division, and finally addition/subtraction. This is the key to correctly calculating each term in the given expression, as shown in the steps above.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. In such a sequence, each term after the first is created by adding a constant difference to the preceding term. However, the given series is not arithmetic but understanding series in arithmetic terms can still benefit us.In our problem, each term is computed through the algebraic expression \(2 + i - i^2\) before being summed. Normally, in an arithmetic series, the difference between consecutive terms is constant, making it predictable. Yet, this exercise involves non-linear changes due to the \(-i^2\) term, which perturbs the steady progression of arithmetic sequences.Despite the difference in conceptual nature, once the terms are generated, they can be summed similarly to an arithmetic series. Each subsequent term might not exhibit a linear growth or decay, but they collectively contribute to a total sum which we calculate step-by-step.

Summation Notation

Summation notation is a concise way to represent the sum of a series. It uses the Greek capital letter sigma (\(\Sigma\)). This notation saves space and clarifies the range of terms being added together. In this exercise, the series is expressed as \(\sum_{i=1}^{6}(2 + i - i^2)\), which means we compute and add terms from \(i = 1\) to \(i = 6\). The key elements include:

• The lower limit of summation, \(i = 1\), denotes where we start.
• The upper limit, \(i = 6\), shows where we finish.
• The expression \(2 + i - i^2\), which is evaluated at each integer value of \(i\) within the given range.

Summation notation enables us to efficiently define and manage problem scopes, especially in complex mathematical studies. It allows focusing on the core calculation task, eliminating the distraction of repetitive additions. This organized approach underpins much of calculus, probability, and statistical theory.