Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
$\sum_{i = 1}^{100} 4 i$

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Identify the arithmetic sequence given by the general term: $a_i = 4i$, where $i$ ranges from 1 to 100.

Write out the first three terms by substituting $i = 1, 2, 3$ into the formula: $a_1 = 4(1)$, $a_2 = 4(2)$, and $a_3 = 4(3)$.

Find the last term by substituting $i = 100$ into the formula: $a_{100} = 4(100)$.

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

Substitute $n = 100$, the first term $a_1$, and the last term $a_{100}$ into the sum formula to express the sum $S_{100}$.

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주요 개념

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Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. For example, in the sequence 4, 8, 12, ..., each term increases by 4. Understanding this helps identify the pattern and calculate specific terms.

General Term of an Arithmetic Sequence

The nth term of an arithmetic sequence can be found using the formula a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. This formula allows you to find any term in the sequence without listing all previous terms.

Sum of the First n Terms of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence is given by S_n = n/2 (a_1 + a_n), where a_1 is the first term and a_n is the nth term. This formula efficiently calculates the total sum without adding each term individually.

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.∑i=11004i\(\sum\)_{i=1}^{100} 4i