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Chapter 7: Problem 1

Evaluate the arithmetic series. $$ 1+2+3+\cdots+98+99+100 $$

Short Answer

Expert verified
The sum of the arithmetic series from 1 to 100 is 5050, calculated using the formula \(S_n = \frac{n(a_1 + a_n)}{2}\), with \(n = 100\), \(a_1 = 1\), and \(a_n = 100\).

Step by step solution

01

Write down the values of n, a_1, and a_n

We are given the arithmetic series: $$ 1+2+3+\cdots+98+99+100 $$ The number of terms \(n = 100\), the first term \(a_1 = 1\), and the last term \(a_n = 100\).
02

Apply the arithmetic series formula

To evaluate the sum of the arithmetic series, we use the formula for the sum of an arithmetic series: $$ S_n = \frac{n(a_1 + a_n)}{2} $$ Now, substitute the values of n, a_1, and a_n in the formula: $$ S_{100} = \frac{100(1 + 100)}{2} $$
03

Evaluate the sum

With the values substituted, let's now evaluate the sum of the series: $$ S_{100} = \frac{100(101)}{2} $$ $$ S_{100} = \frac{10100}{2} $$ $$ S_{100} = 5050 $$ So, the sum of the arithmetic series from 1 to 100 is 5050.

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