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Chapter 15: Problem 44

Evaluate each series. \sum_{i=1}^{6}\left(3 i^{2}-4 i\right)

Short Answer

Expert verified
The sum of the given series \(\sum_{i=1}^{6}\left(3 i^{2}-4 i\right)\) is 189.

Step by step solution

01

Find the terms of the series

We need to calculate the terms of the series by plugging in the values of \(i\) from 1 to 6 into the general term (\(3i^2 - 4i\)). 1. For \(i=1\) : \(3(1)^2 - 4(1) = 3 - 4 = -1\) 2. For \(i=2\) : \(3(2)^2 - 4(2) = 12 - 8 = 4\) 3. For \(i=3\) : \(3(3)^2 - 4(3) = 27 - 12 = 15\) 4. For \(i=4\) : \(3(4)^2 - 4(4) = 48 - 16 = 32\) 5. For \(i=5\) : \(3(5)^2 - 4(5) = 75 - 20 = 55\) 6. For \(i=6\) : \(3(6)^2 - 4(6) = 108 - 24 = 84\) So the terms of the series are -1, 4, 15, 32, 55, and 84.
02

Calculate the sum of the series

Now, we have found all the terms of the series, and we can simply add them up to find the sum of the series. \(\sum_{i=1}^{6}\left(3 i^{2}-4 i\right) = (-1) + 4 + 15 + 32 + 55 + 84 = 189\) Therefore, the sum of the given series is 189.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Summation Notation
Summation notation is a mathematical shorthand way of expressing the sum of a sequence of numbers. It's commonly used when dealing with series or when the same operation needs to be repeated for a list of numbers. The notation
  • 鈭� (the Greek letter Sigma) represents 'sum'.
  • The index, like i, is called the 'index of summation' and indicates the position within the sequence.
  • The lower and upper limits of summation (like 1 to 6) specify the range of values that i takes.
  • The expression next to the Sigma symbol, like \(3i^2 - 4i\), indicates the function or terms to be summed.

Summation notation provides a convenient and concise way to represent the sum of many terms, especially in a series or sequence.
It helps summarize lengthy arithmetic additions into a compact form, making complex calculations more manageable.
Polynomial Series Demystified
A polynomial series involves sums of several polynomial expressions. Each expression in such a series is based on a variable raised to a power.
In our example, the polynomial expressed is \(3i^2 - 4i\).
A few key points to remember about polynomials in a series:
  • Polynomial expressions are algebraic, involving terms like \(a_ni^n+a_{n-1}i^{n-1}+...+a_1i+a_0\).
  • These can be extended and summed over a specified range of i values.
  • Higher degree terms are multiplied by larger coefficients, affecting the overall outcome significantly.

When we evaluate each term by substituting the values, we simplify step by step, ensuring we follow correct order of operations (exponents, multiplication, and addition).
Adding the evaluated terms gives us the total value of the polynomial series.
Exploring Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations (like addition or multiplication).
In the context of our series, \(3i^2 - 4i\) is an algebraic expression involving:
  • Terms: Composed of variables and coefficients. Here, the terms are \(3i^2\) and \(-4i\).
  • Like terms: In a single expression, terms that have the same variable and exponent can be combined. \(3i^2\) and \(-4i\) are unlike terms and cannot be combined as they differ in powers.
  • Coefficient: The number in front of the variable, indicating how many times the variable is being multiplied. Here, 3 and -4 are coefficients respectively.

Understanding the structure of algebraic expressions helps break down complicated problems into simpler parts. This breakdown is essential when each term must be evaluated independently in a series.
Grasping these concepts ensures we can more efficiently handle, simplify, and compute expressions during series evaluation.

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Most popular questions from this chapter

Find the indicated term of each binomial expansion. (w+1)^{15} ; \text { tenth term }

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Find the sum of the terms of the infinite geometric sequence, if possible. $$\frac{7}{2}, \frac{7}{4}, \frac{7}{8}, \frac{7}{16}, \dots$$

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$7,28,112,448,1792,7168,28672$$

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