/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 Find the sum for each series. $$... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 67

Find the sum for each series. $$\sum_{i=1}^{4}\left(3 i^{3}+2 i-4\right)$$

Short Answer

Expert verified
The sum is 304.

Step by step solution

01

Understand the Series Expression

The problem asks us to find the sum of the expression \(3i^3 + 2i - 4\) from \(i = 1\) to \(i = 4\). This means we need to substitute the values \(i = 1\), \(i = 2\), \(i = 3\), and \(i = 4\) into the expression, evaluate them, and then sum the results.
02

Substitute i = 1

Substitute \(i = 1\) into the expression: \(3(1)^3 + 2(1) - 4 = 3 + 2 - 4 = 1\).
03

Substitute i = 2

Substitute \(i = 2\) into the expression: \(3(2)^3 + 2(2) - 4 = 3(8) + 4 - 4 = 24 + 4 - 4 = 24\).
04

Substitute i = 3

Substitute \(i = 3\) into the expression: \(3(3)^3 + 2(3) - 4 = 3(27) + 6 - 4 = 81 + 6 - 4 = 83\).
05

Substitute i = 4

Substitute \(i = 4\) into the expression: \(3(4)^3 + 2(4) - 4 = 3(64) + 8 - 4 = 192 + 8 - 4 = 196\).
06

Sum the Results

Now add the results from each substitution: \(1 + 24 + 83 + 196 = 304\).

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Key Concepts

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

Polynomial Expression
A polynomial expression consists of variables raised to positive integer powers and their coefficients, combined using addition, subtraction, and multiplication. In our example, the expression \(3i^3 + 2i - 4\) is a polynomial with the variable \(i\). It has three terms:
  • \(3i^3\): a cubic term, where \(3\) is the coefficient
  • \(2i\): a linear term, with \(2\) as the coefficient
  • \(-4\): a constant term, meaning it remains the same regardless of \(i\)
Polynomials can take many forms and can best be understood as a simple collection of these terms that fit neatly together under the hood of algebraic rules. They are foundational in mathematics and appear across many domains of math and its applications.
Sigma Notation
Sigma notation is a way to represent the sum of a series of terms that follow a pattern. It's succinct and powerful, especially for complex expressions with many terms. In the original problem, the sigma notation \(\sum_{i=1}^{4}\left(3i^3 + 2i - 4\right)\) tells us:
  • \(\Sigma\) symbolizes the sum.
  • \(i=1\) denotes the starting point for the variable \(i\).
  • \(4\) is the ending point for \(i\).
  • \(3i^3 + 2i - 4\) is the expression whose terms we sum by changing the variable \(i\) from 1 to 4.
This notation is a concise way to convey that each integer value \(i\) within the specified range is plugged into the polynomial, calculated, and then all results are summed up sequentially. It is extremely useful for handling sequences and sums in a formatted and clear manner.
Algebraic Substitution
Algebraic substitution is a straightforward process of replacing a variable in an expression with a number to evaluate the expression. For the series \(\sum_{i=1}^{4}(3i^3 + 2i - 4)\), substitution is the first critical step.
To substitute:
  • Take the variable \(i\).
  • Replace it with each integer value from 1 to 4 in the expression \(3i^3 + 2i - 4\).
  • Calculate each individual expression's value.
For instance:
  • Substitute \(i = 1\) results in \(3(1)^3 + 2(1) - 4 = 1\).
  • Substitute \(i = 2\) gives \(3(2)^3 + 2(2) - 4 = 24\).
  • Continue similarly for \(i = 3\) and \(i = 4\).
Each substitution simplifies the expression so you can calculate its value, paving the way to find the total sum.

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Most popular questions from this chapter

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. In Exercises 69 and 70 , round to the nearest thousandth. $$a_{n}=8.42 n+36.18$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=-3, d=-4$$

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, n)=1$$

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{i=4}^{9} 3(0.25)^{i}$$

Find each sum that converges. $$\sum_{k=1}^{\infty} 3^{-k}$$
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