Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $1^{2} + 2^{2} + 3^{2} + \dots + 15^{2}$

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Identify the pattern in the sum: the terms are squares of consecutive integers starting from 1 up to 15, i.e., 1^2, 2^2, 3^2, ..., 15^2.

Recognize that the index of summation, denoted by $ i $, will represent each integer in the sequence from 1 to 15.

Write the general term of the sum using the index $ i $, which is $ i^2 $ since each term is the square of $ i $.

Set the lower limit of summation to 1, as specified, and the upper limit to 15, the last term in the sum.

Express the entire sum in summation notation as: \[ \sum_{i=1}^{15} i^2 \]

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Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). 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 the sum a_1 + a_2 + ... + a_n.

Index of Summation

The index of summation is a variable, often i, that represents each term's position in the sequence being summed. It starts at the lower limit and increments by 1 until it reaches the upper limit. This index helps define the general term in the summation expression.

Expressing Sums of Powers

When summing powers of integers, such as squares, the general term is written as i raised to the power, e.g., i^2. Expressing sums like 1^2 + 2^2 + ... + 15^2 in summation notation involves writing ∑ from i=1 to 15 of i^2, which compactly represents the entire sum.

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 12+22+32+⋯+1521^2+2^2+3^2+⋯+ 15^2