Question

A(n) _______ is a sum of numbers that accumulates with each iteration of a loop.

Short answer

Answer: Cumulative sum

Step-by-step solution

Define the given term

We are given a term that represents a sum of numbers that accumulates with each iteration of a loop.

Analyze possible terms

Some terms related to summations in programming are: 1. Summation 2. Series 3. Cumulative sum 4. Running total

Identify the best fit term

The term that best fits the given context is 'Cumulative sum' because it represents the partial sums from the sequence of numbers being added at each iteration of a loop.

Answer the question

A(n) Cumulative sum is a sum of numbers that accumulates with each iteration of a loop.

Key concepts

Summation

Summation is the process of adding a sequence of numbers together to get their total. It is a fundamental concept not just in mathematics but also in computing and data analysis.
In mathematical notation, summation is often represented by the sigma symbol (\(\Sigma\)). An example of a summation could be:

• Sum of first 5 natural numbers: \(1 + 2 + 3 + 4 + 5\).

For computational purposes, summation is crucial when performing repetitive additions in code. This is typically seen in algorithms that require aggregation of data or when implementing specific patterns such as arithmetic series.
In the context of programming, summation plays a crucial role in calculating totals and averages, both of which can be especially important in fields such as statistics and economics.

Loop Iteration

Loop iteration is a core part of programming where a block of code executes repeatedly. Loops can be used to perform repetitive tasks efficiently, and they come in different types such as 'for', 'while', and 'do-while'.
Each execution within a loop is termed as an iteration. This concept is fundamental when working with accumulative operations like summation and is especially handy in cases of dealing with collections of data.

• For loops are used when the number of iterations is known beforehand.
• While loops are used when the number of iterations depends on a condition being met during execution.

In loop iterations involving cumulative sums, each iteration adds the next element in a sequence to the current total, effectively constructing a running total. This is the essence of how cumulative sums are generated in programming.

Partial Sums

Partial sums are the totals obtained from adding consecutive elements of a sequence up to a certain point. They are particularly useful for evaluating patterns in sequences and understanding different properties of mathematical series.
In mathematical terms, a partial sum of a series \(a_1, a_2, a_3, \ldots, a_n\) is represented as \(S_k = a_1 + a_2 + \cdots + a_k\). Partial sums are integral in understanding convergent series and identifying patterns or results after a certain number of terms.

• Can visualize peak values and trends in data streams.
• Essential for analyzing time series data effectively.

In programming, calculating partial sums often involves loops where each step accumulates the next value into a running total, which is exactly the concept behind cumulative sums. Understanding partial sums helps in dissecting how values build up sequentially, providing insights into data progression.