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Chapter 6: Problem 8

Evaluate the arithmetic series. $$ \sum_{m=1}^{75}(2+3 m) $$

Short Answer

Expert verified
The sum of the given arithmetic series, \(\sum_{m=1}^{75}(2+3 m)\), is 8700, using the formula for the sum of an arithmetic series \(S_n = \frac{n}{2}(a + l)\), where \(n = 75\), \(a = 5\), and \(l = 227\).

Step by step solution

01

Find the number of terms (n)

From the given series, we can see that it starts from m=1 and goes up to m=75. Hence, the number of terms in the series is \(n = 75\).
02

Identify the common difference (d)

In the series, we can see that the general term is given by \(T_m = 2+ 3m\). If we find the difference between subsequent terms, we can find the common difference (d). If we define \(T_2\) as the second term and \(T_1\) as the first term, we can find the common difference as, \(d = T_2 - T_1 \) which is equivalent to \(d = (2+3(2)) - (2+3(1)) = 5 - 2 = 3\). So, the common difference (d) in the given series is 3.
03

Find the first term (a) and the last term (l)

Now, we identify the first term (a) and the last term (l) of the given series. The first term is given by \(a = 2 + 3(1) = 5\). The last term is given by \(l = 2 + 3(75) = 2 + 225 = 227\).
04

Use the formula for the sum of an arithmetic series

The sum (S) of an arithmetic series can be calculated using the following 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, and l is the last term. Substitute the values we've found into the formula: \[S_{75} = \frac{75}{2}(5 + 227)\]
05

Calculate the sum of the series

Now, to find the sum of the series, perform the operations in the formula: \[S_{75} = \frac{75}{2}(232) = 37.5 * 232 = 8700\] So, the sum of the given arithmetic series is 8700.

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Key Concepts

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

Common Difference
The common difference in an arithmetic series is the constant amount that separates successive terms. This means that each term after the first can be found by adding this constant to the previous term. In the equation for our series, \(T_m = 2 + 3m\), we observe that the terms increase by a regular increment. To determine the common difference (\
Sum 0f Series Formula
The sum of an arithmetic series is a critical calculation, enabling us to find the total of all terms without adding each one separately. We use the formula: \[ S_n = \frac{n}{2}(a + l) \] where:- \(S_n\) represents the sum of the series. - n is the number of terms. - a signifies the first term. - l stands for the las 0f terms, even if they are - consistent increases in 0unts or detail.This ease 0f use makes the formula especially valuable in comparing the total value of complex series.
Number of Terms
In an arithmetic series, knowing the number of terms is essential for calculating the sum of the series. The number of terms can be determined by the sequence provided. For example, if the series begins with a first term, denoted as \(m = 1\), and ends with a last term, \(m = 75\), we determine there are 75 terms. There is no need for additional calculations. It's straightforward: the given range from start to finish indicates how many terms are in the series.
  • Sequential counting: Simply count from 1 to 75, which confirms there are 75 terms.
  • Understanding position: The position of the first and last terms gives the total number of terms.
This understanding is crucial when applying formulas to find the sum or any other calculation involving the entire series.

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