Problem 19 - Calculus

Question

Evaluate \(\sum_{i=1}^{10}\left(6 i^{2}-8 i\right)\).

Step-by-Step Solution

Answer

The sum is 1870.

Step 1: Understand the Summation Notation

The summation notation \( \sum\limits_{i=1}^{10}(6i^2 - 8i) \) indicates that we need to evaluate the expression \( 6i^2 - 8i \) for each integer \( i \) from 1 to 10, and then add all these values together.

Step 2: Evaluate the Expression for Each Integer

Calculate \( 6i^2 - 8i \) for each \( i \) from 1 to 10:- For \( i = 1 \): \( 6(1)^2 - 8(1) = 6 - 8 = -2 \)- For \( i = 2 \): \( 6(2)^2 - 8(2) = 24 - 16 = 8 \)- For \( i = 3 \): \( 6(3)^2 - 8(3) = 54 - 24 = 30 \)- For \( i = 4 \): \( 6(4)^2 - 8(4) = 96 - 32 = 64 \)- For \( i = 5 \): \( 6(5)^2 - 8(5) = 150 - 40 = 110 \)- For \( i = 6 \): \( 6(6)^2 - 8(6) = 216 - 48 = 168 \)- For \( i = 7 \): \( 6(7)^2 - 8(7) = 294 - 56 = 238 \)- For \( i = 8 \): \( 6(8)^2 - 8(8) = 384 - 64 = 320 \)- For \( i = 9 \): \( 6(9)^2 - 8(9) = 486 - 72 = 414 \)- For \( i = 10 \): \( 6(10)^2 - 8(10) = 600 - 80 = 520 \)

Step 3: Sum the Evaluated Values

Add all the evaluated values together:- \( -2 + 8 + 30 + 64 + 110 + 168 + 238 + 320 + 414 + 520 \).Perform the addition:- Step 1: \( -2 + 8 = 6 \)- Step 2: \( 6 + 30 = 36 \)- Step 3: \( 36 + 64 = 100 \)- Step 4: \( 100 + 110 = 210 \)- Step 5: \( 210 + 168 = 378 \)- Step 6: \( 378 + 238 = 616 \)- Step 7: \( 616 + 320 = 936 \)- Step 8: \( 936 + 414 = 1350 \)- Step 9: \( 1350 + 520 = 1870 \)

Key Concepts

Summation Notation

Summation notation is a powerful tool used in mathematics to represent the sum of a sequence of terms. This allows us to write long sums in a compact form, which is very useful for both practical computations and theoretical work. The symbol \( \sum \) is the Greek capital letter sigma and is used to denote summation. For example, the expression \( \sum\limits_{i=1}^{10}(6i^2 - 8i) \) means we are summing the values of the expression \( 6i^2 - 8i \) as \( i \) ranges from 1 to 10.Some important points about summation notation:

• Index variable: The variable \( i \) is known as the index of summation and usually takes on integer values in the specified range.
• Limits of summation: The numbers below and above the \( \sum \) symbol are the lower and upper limits, respectively. They specify the range of the index variable.
• Expression: The formula to the right of the \( \sum \) sign is the expression being summed.

Breaking it down, summation notation greatly simplifies complex calculations, especially when working with large sequences.

Evaluating Expressions

Evaluating expressions involves calculating the result of a mathematical expression by substituting values for any variables. In this context, we substitute different integer values for \( i \), one at a time, in the expression \( 6i^2 - 8i \).To evaluate the expression:

• Substitute: Plug in the integer values for \( i \), ranging from 1 to 10 in this example.
• Calculate: Perform the arithmetic operations in the expression for each value of \( i \). For instance:
  • When \( i = 1 \), the expression becomes \( 6(1)^2 - 8(1) = 6 - 8 = -2 \).
  • As another example, when \( i = 2 \), it becomes \( 6(2)^2 - 8(2) = 24 - 16 = 8 \).
• Repeat: Continue this process up to the upper limit of the summation.

Evaluating each term precisely ensures the accuracy of the final sum.

Integer Sequences

Integer sequences are ordered lists of integers, which follow a specific rule or pattern. In mathematics, sequences serve as fundamental tools for understanding patterns and making calculations.With regard to summation, a sequence is generated by applying a fixed formula to each integer within a specified range. Using our exercise as an example, we generate the sequence:

• \( i = 1: -2 \)
• \( i = 2: 8 \)
• \( i = 3: 30 \)
• \( i = 4: 64 \)
• \( i = 5: 110 \)
• \( i = 6: 168 \)
• \( i = 7: 238 \)
• \( i = 8: 320 \)
• \( i = 9: 414 \)
• \( i = 10: 520 \)

Each term is derived by replacing \( i \) in the expression \( 6i^2 - 8i \) and evaluating it. Recognizing and working with integer sequences is essential for developing skills to solve complex mathematical problems.