/*! 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 1 \(\sum_{i=1}^{6}(3 i-2)\)... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 1

\(\sum_{i=1}^{6}(3 i-2)\)

Short Answer

Expert verified
The sum is 51.

Step by step solution

01

Understand the Summation Notation

The summation notation \(\backslash sum_{{i=1}}^{{6}}(3 i - 2)\) indicates that we need to evaluate the expression \((3i - 2)\) for values of \(i\) ranging from 1 to 6 and then sum all these values.
02

Evaluate Each Term in the Series

Calculate the value of \((3i - 2)\) for each \(i\) from 1 to 6.\When \(i = 1\): \(3 \times 1 - 2 = 1\).\When \(i = 2\): \(3 \times 2 - 2 = 4\).\When \(i = 3\): \(3 \times 3 - 2 = 7\).\When \(i = 4\): \(3 \times 4 - 2 = 10\).\When \(i = 5\): \(3 \times 5 - 2 = 13\).\When \(i = 6\): \(3 \times 6 - 2 = 16\).
03

Sum the Evaluated Terms

Add up all the values calculated in Step 2.\(1 + 4 + 7 + 10 + 13 + 16 = 51\)

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Key Concepts

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

series evaluation
Evaluating a series means calculating the sum of a sequence of terms. In our problem, the summation notation \(\backslash sum_{{i=1}}^{{6}}(3 i - 2)\) tells us that we need to evaluate the expression \((3i - 2)\) for values of \(i\) ranging from 1 to 6 and then add up these values.
To perform series evaluation effectively:
  • Firstly, identify the expression to be summed, which in our case is \((3i - 2)\).
  • Secondly, determine the range of values for the index \(i\), here from 1 to 6.
You calculate \(3i - 2\) for each value of \(i\) within the given range and sum up these results. This step-by-step approach helps in breaking down the problem into manageable parts, making it easier to understand and solve.
arithmetic series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. The series \(\backslash sum_{i=1}^{6}(3i - 2)\) can be viewed as an arithmetic series.
Here鈥檚 why:
  • Each term in the sequence is derived from the previous term by adding a constant value (which in this case is 3).
  • We start with 1, then add 3 to get 4, add 3 again to get 7, and this continues.
In general, we can describe an arithmetic series by its first term \(a\) and common difference \(d\). For our example:
  • The first term \(a = 1\).
  • The common difference \(d = 3\).
Understanding that our series fits the definition of an arithmetic series helps recognize patterns and use known formulas to evaluate the sum more efficiently.
summation formula
The summation formula is a tool used to find the sum of terms in a sequence or series. For an arithmetic series, the sum can be calculated using the formula:
\[ S_n = \frac{n}{2} (a + l) \]
where:
  • \(S_n\) is the sum of the series.
  • \(n\) is the number of terms.
  • \(a\) is the first term.
  • \(l\) is the last term.
For our specific example, where the series is
\([-1] + 4 + 7 + 10 + 13 + 16\):
  • The first term \(a = 1\).
  • The last term \(l = 16\).
  • The number of terms \(n = 6\).
Plugging these values into the formula, we get: \[ S_6 = \frac{6}{2} (1 + 16) = 3 \times 17 = 51 \] This formula provides a quick and easy way to find the sum of an arithmetic series without having to add each term individually.

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