Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)

Sum of the First n Natural Numbers (∑r)

The symbol ∑ stands for “sum of” and r represents the terms in a sequence, starting from r=1 up to a given number n.
The sum of the first n natural numbers, denoted as ∑r, is calculated using the formula n(n+1)/2.
For example, the sum of the first 10 natural numbers (which are 1, 2, 3, … , 10) with n=10 is given by 10(10+1)/2 = 55.
This formula is derived from the mathematical pattern observed in adding sequences of natural numbers.

When considering constant sequences where each term is a constant value a, the sum is simply n times the constant.
The sum of the first n terms of a constant sequence, denoted as ∑a, is calculated using the formula an.
That’s because you’re simply adding the constant a n times.
For example, for a sequence of constant 5 for n=3 (i.e., 5, 5, 5), the sum is 3*5 = 15.

For a sequence where each term is ar+b, with a and b as constants and r as the term number, you can separate the sum of terms into two separate sums: a∑r + b∑1.
The result for ∑(ar+b) from r=1 to n is given by a∑r + bn.
Here ∑r is the sum of the first n natural numbers and ∑1 is the sum of n number of 1s, which is n.
In practical terms, this means that we sum the elements of the ar series, and then add to that the sum of the b series, which is just b added n times.
For example, for a=3, b=2, and n=4 (i.e., the sequence 5, 8, 11, 14), ∑(ar+b) is given by 3(4(4+1)/2) + 2*4 = 30 + 8 = 38.