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Chapter 12: Problem 52

Expand the partial sum and find its value. \(\sum_{i=1}^{6}(3 i-2)\)

Short Answer

Expert verified
The value of the sum is 51.

Step by step solution

01

Identify the Given Sum

The given sum is \(\sum_{i=1}^{6}(3i-2)\). This is a summation notation where each term is given by the formula \(3i-2\) and we are summing from \(i=1\) to \(i=6\).
02

Calculate Each Term in the Sum

To expand the sum, calculate the value of the expression \(3i-2\) for each \(i\) from 1 to 6: \ - For \(i = 1, (3 \times 1) - 2 = 3 - 2 = 1\) - For \(i = 2, (3 \times 2) - 2 = 6 - 2 = 4\) - For \(i = 3, (3 \times 3) - 2 = 9 - 2 = 7\) - For \(i = 4, (3 \times 4) - 2 = 12 - 2 = 10\) - For \(i = 5, (3 \times 5) - 2 = 15 - 2 = 13\) - For \(i = 6, (3 \times 6) - 2 = 18 - 2 = 16\)
03

Write the Expanded Sum

The expanded form of the sum is \(1 + 4 + 7 + 10 + 13 + 16\).
04

Add the Terms

Add the values calculated in the previous step: \[1 + 4 + 7 + 10 + 13 + 16 = 51\]

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Key Concepts

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

Partial Sum
A partial sum is the sum of the first few terms of a series, not the entire series. In this exercise, we deal with a partial sum because we only sum up the terms from i=1 to i=6. Partial sums are useful when examining the behavior of a series before summing it entirely. By calculating partial sums, we can better understand the relationships within the series. Here, our formula was \(\sum_{i=1}^{6}(3i-2)\). We plugged in values from 1 to 6 and computed the respective terms: 1, 4, 7, 10, 13, and 16.
  • Recognizing a partial sum helps simplify complex problems.
  • Partial sums provide insight into the trends in the data.
  • They are often used in mathematical proofs and analysis.
Series Expansion
Series expansion means expressing a summation notation in its detailed form. This involves breaking down the sigma notation into individual terms. For example, in \(\sum_{i=1}^{6}(3i-2)\), we expanded it to: 1 + 4 + 7 + 10 + 13 + 16. This step-by-step breakdown makes it easier to compute the sum.
Series expansion is particularly helpful for:
  • Identifying patterns in the series.
  • Visualizing and better understanding the terms.
  • Ensuring accuracy by showing each step of the computation.
Arithmetic Series
An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. In our exercise, we have an arithmetic series with a common difference of 3 ((4-1), (7-4), (10-7), (13-10), and (16-13)). Recognizing an arithmetic series allows us to use specific formulas for quick and accurate calculations.
Key properties of an arithmetic series:
  • Consistent difference between consecutive terms.
  • Simple formula for the n-th term: \( {a_n = a_1 + (n-1)d \), where a1 is the first term and d is the common difference.
  • Sum of the first n terms can be found using the formula: \( S_n = \frac{n}{2}(a_1 + a_n) \).
Summation Formula
Summation notation (Sigma notation) is a concise way to represent the sum of a sequence of terms. The general form is \(\sum_{i=a}^{b} f(i)\), where 'a' is the starting value, 'b' is the ending value, and 'f(i)' is the function describing the terms. For our exercise, it was \(\sum_{i=1}^{6}(3i-2)\). Summation formulas are powerful because:
  • They provide an efficient way to compute series.
  • Many mathematical proofs rely on summation formulas.
  • They help streamline and organize complex calculations.
Understanding the summation formula and how to apply it can greatly enhance your ability to solve and comprehend advanced mathematical problems.

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