Arithmetic Sequences - Key Formulas and Proofs

Ever noticed how cinema seats are numbered or how your savings grow each week? That's arithmetic sequences in action - patterns where you add the same amount each time.

The nth term formula is your best mate: a + $n-1$d. Here, 'a' is your starting number (first term) and 'd' is the common difference you keep adding. Want the 10th term? Just plug in n = 10 and you're sorted.

For finding the sum of an arithmetic sequence, use S_n = n/2$a + L$, where L is the last term. This saves you from adding up dozens of terms manually - brilliant for exam time pressure.

The proof shown demonstrates why this formula works by writing the sum forwards and backwards, then adding them together. When you add 2S = n$2a + (n-1)d$, dividing by 2 gives you the standard sum formula. Understanding this proof shows examiners you grasp the underlying maths, not just memorised formulas.

Quick Tip: Always identify 'a' and 'd' first - they're the building blocks for everything else in arithmetic sequences!