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Chapter 14: Problem 57

Use a formula for \(S_{n}\) to evaluate each series. \(\sum_{i=1}^{250} i\)

Short Answer

Expert verified
31375

Step by step solution

01

- Identify the Formula

Recognize that the given series is an arithmetic series. The formula to find the sum of the first n natural numbers is \[S_n = \frac{n(n+1)}{2}\]
02

- Identify the Value of n

In the given series, the upper limit of the summation is 250, so \(n = 250\).
03

- Substitute n into the Formula

Substitute 250 for n in the formula: \[S_{250} = \frac{250(250+1)}{2}\]
04

- Calculate the Sum

Perform the calculation: \[S_{250} = \frac{250 \times 251}{2} = \frac{62750}{2} = 31375\]

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Key Concepts

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

arithmetic series
An arithmetic series is a sum of terms in a sequence where each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is called the common difference. For example, the sequence 1, 2, 3, 4,... forms an arithmetic sequence with a common difference of 1.
An arithmetic series can be written in summation notation, such as \(\sum_{i=1}^{n} a_i\), where 'n' is the number of terms. To find the sum of an arithmetic series, you can use specific formulas depending on the sequence characteristics.
sum formula
The sum formula for an arithmetic series is a useful tool when dealing with sequences. The formula to find the sum of the first 'n' natural numbers is:
\[ S_n = \frac{n(n+1)}{2} \]
Here, \( S_n \) represents the sum of the first 'n' terms, and 'n' is the number of terms.
For example, to find the sum of the first 250 natural numbers, we use the formula:
\[ S_{250} = \frac{250(250+1)}{2} \]
This simplifies the process and helps avoid manually adding each term. By substituting \( n \) with 250, the formula calculates quickly:
\[ S_{250} = \frac{250 \times 251}{2} = 31,375 \]
Thus, the sum of the first 250 natural numbers is 31,375.
summation
Summation notation (\( \sum \)) is a concise way to represent the sum of a sequence of terms. The expression \( \sum_{i=1}^{n} a_i \) tells us to sum all terms \( a_i \) from \( i=1 \) to \( i=n \).
An important aspect of summation is its efficiency in solving arithmetic series problems. For instance, to find \( \sum_{i=1}^{250} i \), the problem can be approached with the use of the sum formula for arithmetic series. Recognizing this simplifies the process:
  • Identify the formula needed (\[ S_n = \frac{n(n+1)}{2} \]).
  • Identify the value of 'n' (which is 250 in this case).
  • Substitute 'n' into the formula.
  • Calculate the sum to find the answer, 31,375.
Using summation notation, combined with the appropriate formulas, makes solving such problems straightforward and efficient.

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Most popular questions from this chapter

Fill in each blank with the correct response. The number of terms in the geometric sequence \(1,2,4, \ldots, 2048\) is ______.

Determine an expression for the general term \(a_{n}\) of each sequence \(7,14,21,28, \ldots\)

Evaluate each expression. $$ 4 ! $$

Write the first five terms of each sequence. $$ a_{n}=-\frac{2}{n^{2}} $$

Find the indicated term for each arithmetic sequence. \(a_{1}=4, d=3 ; \quad a_{25}\)
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