In Exercises 43–54, 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+⋯+ 15^2

Verified step by step guidance

1. Identify the pattern in the series: 1^2, 2^2, 3^2, ..., 15^2.
2. Recognize that each term in the series is a square of an integer starting from 1 up to 15.
3. The general term of the series can be expressed as i^2, where i is the index of summation.
4. Determine the lower and upper limits of the summation. The series starts at 1 and ends at 15, so the limits are from i = 1 to i = 15.
5. Express the series using summation notation: \( \sum_{i=1}^{15} i^2 \).

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It typically uses the Greek letter sigma (Σ) to denote the sum, with limits indicating the starting and ending values of the index. For example, Σ from i=1 to n signifies the sum of terms indexed by i, starting at 1 and ending at n.

Index of Summation

The index of summation is a variable that represents the position of each term in the sequence being summed. In the expression Σ from i=1 to n, 'i' is the index that takes on integer values starting from 1 up to n. This index allows for the systematic addition of terms, such as i^2 in the case of summing squares.

Series of Squares

A series of squares refers to the sum of the squares of consecutive integers. In this context, the expression 1^2 + 2^2 + 3^2 + ... + 15^2 represents the sum of the squares of the first 15 positive integers. This can be expressed in summation notation as Σ from i=1 to 15 of i^2, which simplifies the representation of the sum.