Problem 67 Evaluate each sum using a formul... [FREE SOLUTION]

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Chapter 15: Problem 67

Evaluate each sum using a formula for $S_{n}$. $$\sum_{i=1}^{18}(3 i-11)$$

Short Answer

Expert verified

The short answer is: $\sum_{i=1}^{18}(3 i-11) = 315$.

Step by step solution

Find the first term of the series

Plug i=1 into the formula (3i - 11): $a_1 = 3(1) - 11 = -8$

Find the last term of the series

Plug i=18 into the formula (3i - 11): $a_{18} = 3(18) - 11 = 54 - 11 = 43$

Apply the formula for the sum of an arithmetic series

Plugging n=18, $a_1=-8$, and $a_{18}=43$ into the sum formula $S_n = \frac{n}{2} (a_1 + a_n)$, we get: \[S_{18} = \frac{18}{2}(-8 + 43) = 9 \times 35 = 315\] So, $\sum_{i=1}^{18}(3 i-11) = 315$.

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Key Concepts

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

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is derived by adding a consistent value, known as the common difference, to the previous term. This sequence forms a linear pattern. For the series\[ a_1, a_2, a_3, \ldots, a_n \],
the common difference $d$ is given by $d = a_2 - a_1$.

To better understand how this sequence works, consider the example:

First term ($a_1$): -8
Second term ($a_2$): -5
Common difference ($d$): 3

So, each term is formed by adding $3$ to the previous term: -8, -5, -2, ... .
This consistent addition of the common difference makes arithmetic progressions predictable and easy to handle.

Series Evaluation

Evaluating a series involves finding the sum of its terms. In the context of arithmetic progressions, this is often accomplished by determining the sum from the first term to a specified last term, using established formulas. The example provided shows a series:

First term ($a_1$): -8
Last term ($a_n$): 43

The objective here is to determine the total sum, which involves using a specific summation formula for arithmetic series.

This process is essential for understanding how sequences add up and is broadly applicable in real-world scenarios where one needs to find the aggregate of regularly spaced data points.

Summation Formula

The summation formula for an arithmetic series is key to efficiently finding the sum without manually adding each term. The formula is expressed as:\[ S_n = \frac{n}{2} (a_1 + a_n) \]
where:

$S_n$ is the sum of the series,
$n$ is the number of terms,
$a_1$ is the first term, and
$a_n$ is the last term.

In the provided solution, plugging in $n = 18$, $a_1 = -8$, and $a_{18} = 43$ into this equation leads to:\[ S_{18} = \frac{18}{2}(-8 + 43) = 9 \times 35 = 315 \].

This elegant equation simplifies calculations greatly, making it easy to derive the sum of large sequences with minimal computations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and multiplication). In the context of arithmetic progressions, these expressions are used to define sequences and find specific terms or their sums.

The expression $3i - 11$ in the original problem represents each term in the sequence. By substituting different values of $i$, one can generate successive terms:

For $i=1$, term is $3(1) - 11 = -8$
For $i=18$, term is $3(18) - 11 = 43$

These expressions are crucial for translating real-world problems into mathematical language, allowing us to use algebraic techniques to analyze and solve them, as demonstrated in the solution provided.

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91影视

Evaluate each sum using a formula for \(S_{n}\). $$\sum_{i=1}^{18}(3 i-11)$$

Short Answer

Step by step solution

Find the first term of the series

Find the last term of the series

Apply the formula for the sum of an arithmetic series

Key Concepts

Arithmetic Progression

Series Evaluation

Summation Formula

Algebraic Expressions

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