Sum of arithmetic progression
When it comes to such a parameter as the sum of arithmetic progression, it always implies the sum of the first terms of the arithmetic progression or the sum of the terms of the progression from k to n, i.e., the number of terms taken for the sum is strictly limited within the conditions set. Otherwise, the task will have no solution, as the entire numerical sequence of the arithmetic progression begins with a specific number - the first term a1, and continues indefinitely.
It is believed that the formula for the sum of arithmetic progression was discovered by Gauss as a quick and accurate way to calculate the sum of numbers in a specific sequence. He noticed that such a progression is symmetrical, meaning the sum of symmetrically arranged terms from the beginning and end of the progression is constant for the given series.
Accordingly, he found this sum and multiplied it by half of the total number of numbers in the sequence involved in the sum calculation. Thus, the formula for the sum of arithmetic progression was derived
Example. Suppose the condition is given: "Find the sum of the first ten (10) terms of the arithmetic progression". For this, the following data is needed: the difference of the progression and its first term. If the problem provides any n term of the arithmetic progression instead of the first, then you first need to use the section where the formula for finding the first term of the progression is presented, and find it. Then the initial data is entered into the calculator, and it performs calculations by adding the first and tenth terms and multiplying the resulting sum by half of the total number of added terms – by 5. Similarly, if you need to find the sum of the first six terms or any other quantity.
In case it is necessary to find the sum of the terms of the arithmetic progression starting not with the first, but with the fifth term, for example, then the arithmetic mean remains the same, and the total number of terms is taken as the increased by one difference between the ordinal numbers of the taken terms.
