Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+2+3+⋯+ 30

검증된 단계별 안내

Identify the pattern in the sum: the terms are consecutive integers starting from 1 up to 30.

Recognize that the sum can be expressed as the sum of \( i \) where \( i \) takes on integer values from 1 to 30.

Write the summation notation using the sigma symbol \( \sum \), the index of summation \( i \), the lower limit 1, and the upper limit 30.

Express the sum as \( \sum_{i=1}^{30} i \), which means adding all integers \( i \) starting at 1 and ending at 30.

This notation compactly represents the original sum \( 1 + 2 + 3 + \cdots + 30 \) in a concise mathematical form.

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주요 개념

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Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the Greek letter sigma (∑). It includes an index of summation, lower and upper limits, and the general term to be summed. For example, ∑ from i=1 to n of a_i represents adding terms a_1 through a_n.

Index of Summation

The index of summation is a variable, often i, that represents the position of each term in the sum. It starts at the lower limit and increments by one until it reaches the upper limit. This index helps define each term in the sequence being summed.

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. For example, 1 + 2 + 3 + ... + 30 is an arithmetic series with a common difference of 1. Recognizing this helps in expressing the sum using summation notation.