MP Board · Class 10 · Mathematics · Arithmetic ProgressionsDerive the formula for the sum of the first n terms of an Arithmetic Progression (AP) and discuss its importance with at least two real‑world applications.

Derivation of the Sum Formula

1. Definition of an AP

• An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant called the common difference (d) to the preceding term.
• General term: (a_n = a + (n-1)d), where (a) is the first term.

2. Writing the series forward and backward

• Sum of the first (n) terms: [S_n = a + (a+d) + (a+2d) + \dots + [a+(n-1)d]]
• Write the same series in reverse order: [S_n = [a+(n-1)d] + [a+(n-2)d] + \dots + a]

3. Adding the two expressions term‑wise

• Adding the corresponding terms of the two rows gives the same result for each pair: [S_n + S_n = [2a + (n-1)d] + [2a + (n-1)d] + \dots + [2a + (n-1)d]]
• There are (n) such identical pairs, therefore: [2S_n = n[2a + (n-1)d]]

4. Isolating (S_n)

• Divide both sides by 2: [S_n = \frac{n}{2}\bigl[2a + (n-1)d\bigr]]
• This is the required formula for the sum of the first (n) terms of an AP.

Real‑World Applications

• Finance – Fixed‑Rate Installments: When a borrower repays a loan in equal instalments with a fixed increase each period (e.g., a salary increment plan), the total amount paid over (n) periods can be calculated using the AP sum formula.
• Construction – Staggered Bricklaying: In a wall where each successive layer of bricks is set back by a constant distance, the total horizontal offset after (n) layers is obtained by the AP sum, aiding architects in precise planning.

Conclusion\nThe sum formula (S_n = \frac{n}{2}[2a+(n-1)d]) is derived by a simple pairing technique and is indispensable in many practical contexts, ranging from financial calculations to engineering designs, wherever quantities change uniformly.