Arithmetic Sequences Explained Simply

An arithmetic series is the sum of the terms in an arithmetic sequence.The sum of the first n terms of an arithmetic sequence can be calculated using this formula: S-n equals n divided by 2, multiplied by the sum of the first term a-1 and the last term a-n.In this formula, S-n represents the sum of the first n terms. a-1 is the first term, a-n is the nth or last term, and n is the number of terms we're adding.Alternatively, we can use this formula: S-n equals n divided by 2, multiplied by the quantity 2 times a-1 plus n minus 1 times d, where d is the common difference.Let's work through an example. We'll find the sum of the first 10 terms of an arithmetic sequence with a first term of 3 and a common difference of 4.Let's visualize our sequence. Each term increases by the common difference of 4.Let's calculate the sum using our formula. We have n equals 10, a-1 equals 3, and d equals 4.Substituting our values into the formula, we get S-10 equals 10 divided by 2, multiplied by the quantity 2 times 3 plus 10 minus 1 times 4.Simplifying, we get 5 times the quantity 6 plus 9 times 4.9 times 4 equals 36, so we have 5 times the quantity 6 plus 36.Adding inside the brackets, we get 5 times 42.And our final answer is 210.This means the sum of the first 10 terms of our arithmetic sequence is 210. Let's visualize this sum with 210 squares.The formula for the sum of an arithmetic sequence is particularly useful because it saves us from having to add each term individually.Without the formula, we would need to calculate each term and then add them all together, which is time-consuming and prone to errors. Using the formula gives us a quick and accurate result.To summarize, we can find the sum of an arithmetic sequence using either of these formulas, choosing the one that works best with the information we have.