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

Guida verificata passo dopo passo

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: \[ \sum_{i=1}^{15} i^2 \]

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

05:18

Interval Notation

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.

Video consigliato:

03:50

Adding & Subtracting Like Radicals

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.

Video consigliato: