how to understand the summation formula for arithmetic series

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Welcome to understanding arithmetic series! An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant amount called the common difference. Let's look at an example: the sequence 2, 5, 8, 11, 14 has a common difference of 3. The sum of all these terms gives us the arithmetic series. The brilliant insight behind the arithmetic series formula is the pairing method. We write the series both forward and backward, then pair terms from opposite ends. Notice something amazing: each pair always sums to the same value! In our example, 2 plus 14 equals 16, 5 plus 11 equals 16, and 8 plus 8 equals 16. This constant sum is the key to our formula. Now let's derive the formula step by step. We add the forward and backward series together. This gives us 2S equals 5 times 16, which equals 80. Dividing both sides by 2, we get S equals 40. This leads us to the general formula: S sub n equals n over 2 times the quantity a sub 1 plus a sub n. This elegant formula works for any arithmetic series! Sometimes you don't know the last term, but you know the first term and common difference. In this case, we use the alternative formula. Since a sub n equals a sub 1 plus n minus 1 times d, we substitute this into our original formula. This gives us S sub n equals n over 2 times the quantity 2 a sub 1 plus n minus 1 times d. Let's verify with our example: this also gives us 40, confirming both formulas work! To summarize what we've learned: Arithmetic series sum the terms of arithmetic sequences using elegant formulas. The pairing method shows why these formulas work by revealing that opposite terms always sum to the same value. We have two equivalent formulas: one using the first and last terms, and another using the first term and common difference. Both approaches give the same result and can be applied to solve any arithmetic series problem efficiently.