Question

write out the first three terms and the last term of the sequence. then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
\\(\sum_{i=1}^{100} 2i\\)

\\(a_1 = 2\\)

\\(a_2 = 4\\)

\\(a_3 = 6\\)

the last term is 200

the sum of the series is \\(\square\\).

Explanation:

Step1: Recall the arithmetic series sum formula

The formula for the sum of the first \( n \) terms of an arithmetic sequence is \( S_n=\frac{n(a_1 + a_n)}{2} \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the \( n \)-th (last) term.

Step2: Identify the values of \( n \), \( a_1 \), and \( a_n \)

Here, \( n = 100 \) (since we are summing from \( i = 1 \) to \( i = 100 \)), \( a_1 = 2 \), and \( a_{100}=200 \).

Step3: Substitute the values into the formula

Substitute \( n = 100 \), \( a_1 = 2 \), and \( a_{100}=200 \) into the formula \( S_n=\frac{n(a_1 + a_n)}{2} \):

$$ S_{100}=\frac{100\times(2 + 200)}{2} $$

First, calculate the sum inside the parentheses: \( 2+200 = 202 \).
Then, multiply by \( n \): \( 100\times202 = 20200 \).
Finally, divide by \( 2 \): \( \frac{20200}{2}=10100 \).

Answer:

\( 10100 \)