Final answer:
An algorithm to compute the sum of the first n terms of a series can be written using a loop and adding the value of each term to a total variable.
Step-by-step explanation:
An algorithm to compute the sum of the first n terms of a series can be written as follows:
- Initialize a variable total to 0.
- Use a loop to iterate n times:
- Within the loop, calculate the value of each term in the series.
- Add the value of each term to the total.
- At the end of the loop, the variable total will contain the sum of the first n terms of the series.
For example, let's calculate the sum of the first 5 terms of the series where each term is equal to its position squared:
total = 0
For i = 1 to 5:
total = total + i^2
The final value of total will be the sum of the first 5 terms.
The sum of the first n terms of a series, where the expression has been given as n², can be computed by systematically pairing and adjusting terms to reveal that their sum is 2n².
A student has asked for an algorithm to compute the sum of the first n terms of a series. To do this, we start with an understanding that each term in the series has a pattern that can be manipulated mathematically. Considering the expression provided, where the sum of the series is equal to n², we can see that modifying the terms strategically will result in a simple expression to find the sum.
By taking (n - 1) from the last term and adding it to the first term, and then (n - 3) from the penultimate term and adding it to the second term, and so on till we reach terms in the middle of the sequence that are all equal to n, we observe that the sum of each pair of terms is 2n. Since there are n such terms, the sum is calculated as 2n² using this paired approach.
It is instructive to note that this approach applies to a specific type of series where each term is in the form of an arithmetic progression or follows a recognizable pattern that allows for such pairing and simplification.