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Chapter 14: Problem 40

Use a formula for \(S_{n}\) to evaluate each series. $$ \sum_{i=1}^{2000} i $$

Short Answer

Expert verified
The sum is 2001000.

Step by step solution

01

Identify the Series Type

The given series is a sum of the first 2000 natural numbers, which is an arithmetic series where each term increases by a constant value of 1.
02

Recall the Sum Formula for an Arithmetic Series

The formula to find the sum of the first n natural numbers is \[ S_n = \frac{n(n+1)}{2} \].
03

Substitute Values into the Formula

Substitute n = 2000 into the formula: \[ S_{2000} = \frac{2000 \times (2000 + 1)}{2} \].
04

Compute the Sum

Calculate the value inside the formula: \[ S_{2000} = \frac{2000 \times 2001}{2} = \frac{4002000}{2} = 2001000 \].

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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 an arithmetic sequence. An arithmetic sequence is where the difference between consecutive terms is always the same. For example, in the sequence 1, 2, 3, 4, ..., the difference between each number is 1, so it's an arithmetic sequence.
When we sum these terms, we get an arithmetic series. The series can be finite (having a fixed number of terms) or infinite (ongoing forever). In our problem, we are dealing with a finite arithmetic series, summing the first 2000 natural numbers.
Understanding that our series is arithmetic helps us apply the right formulas to find the sum efficiently.
sum of first n natural numbers
The sum of the first n natural numbers is a special case of an arithmetic series. Natural numbers are simply the counting numbers: 1, 2, 3, 4, etc. When we want to add up the first n natural numbers, we are essentially summing an arithmetic series where the first term (a) is 1 and the common difference (d) is also 1.
The sequence of the first n natural numbers is 1, 2, 3, ..., n. The sum of this sequence can be quickly found with a specific formula. Recognizing this pattern allows students to avoid long and tedious addition processes, providing a shortcut to the final sum.
formula for series sum
To find the sum of the first n terms of an arithmetic series, we use the formula:
evaluating series
Evaluating a series, especially an arithmetic series, means finding its total sum. To do this efficiently, we use formulas that simplify the computation.
In our example, we use the formula for the sum of the first n natural numbers: --i-ththththth-nththththth-thththorn-thtocomplete--.-.-thandaxoncanassumes-reduce-series-insimplesorandelearning-findconceptsax-reducehelpsandtheeffortprocess-formulas-suitorcontactup-makes-problemsolve-ofuse

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Most popular questions from this chapter

Find the arithmetic mean for each collection of numbers. $$ 2,1,4,8,3,7,10,8,0 $$

Use the formula for \(a_{n}\) to find the general term of each arithmetic sequence. $$ a_{1}=2, d=5 $$

Write out the first five terms of each sequence. $$ a_{n}=\frac{1}{n^{2}} $$

Solve each problem involving an ordinary annuity. A father opened a savings account for his daughter on her first birthday, depositing \(\$ 1000 .\) Each year on her birthday he deposits another \(\$ 1000,\) making the last deposit on her 21 st birthday. If the account pays \(4.4 \%\) interest compounded annually, how much is in the account at the end of the day on the daughter's 21 st birthday? (IMAGE CANNOT COPY)

If the given sequence is arithmetic, find the common difference d. If the sequence is not arithmetic, say so. $$ 2,-4,6,-8,10,-12, \dots $$
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