Arithmetic progression (AP) — nth term, sum of n terms — derivations

3.3.1 · Maths › Sequences & Series

AP KYA hai?

WHY ise difference se define karte hain? Kyunki "constant difference" hi woh ek aur ekmaatra rule hai jo AP follow karta hai. Ek baar yeh pata chal gaya, toh har term automatically determined ho jaata hai.

nth term derive karna (scratch se)

HOW — ise build up karo:

a_{1} a_{2} a_{3} a_{4} a_{n} = a = a + d = a + 2 d = a + 3 d ⋮ = a + (n - 1) d (0 steps) (1 step) (2 steps) (3 steps) (n - 1 steps)

Sum $S_{n}$ derive karna (Gauss ka trick)

HOW — derivation:

Sum likhо, phir use reverse karo:

S_{n} S_{n} = a + (a + d) + (a + 2 d) + \dots + (a + (n - 1) d) = (a + (n - 1) d) + \dots + (a + 2 d) + (a + d) + a

Dono lines ko column by column add karo. Har column $2 a + (n - 1) d = a_{1} + a_{n}$ deta hai:

2 S_{n} = n equal terms (2 a + (n - 1) d) + \dots + (2 a + (n - 1) d) = n (2 a + (n - 1) d)

2 se divide karo:

WHY doosri form beautiful hai: sum = (number of terms) × (average term). AP mein average sirf $\frac{first + last}{2}$ hota hai kyunki terms evenly spaced hoti hain.

Figure — Arithmetic progression (AP) — nth term, sum of n terms — derivations

$a_{n}$ aur $S_{n}$ ke beech useful links

Worked Examples

Common Mistakes (Steel-manned)

80/20 — vital 20%

Recall Feynman: ek 12-year-old ko explain karo

Socho tum ek aisi stairs chadh rahe ho jahan har step ki height same hai. Tumhari starting jagah $a$ hai; har step tumhe $d$ se upar uthati hai. Step number $n$ pe tumhari height jaanne ke liye, tumne sirf $(n - 1)$ steps chadhe hain (step 1 tak pahunchne ke liye tum chadhe nahi — tum already wahan the!), toh height $= a + (n - 1) d$ . Ab saari stairs ki heights add karne ke liye: unhe shortest se tallest tak line up karo, phir same stairs tallest se shortest tak ek side mein rakh do. Ab har pair ki total height same hai. $n$ pairs hain, har ek $= (first + last)$ , toh double answer $= n (first + last)$ . Ise halve karo aur ho gaya!

Connections

Geometric Progression (GP) — multiply-by-ratio cousin of the add-by- $d$ AP.
Arithmetic Mean — 3-term AP ka middle term $= \frac{a + c}{2}$ .
Sigma Notation — $S_{n} = \sum_{k = 1}^{n} a_{k}$ .
Quadratic Equations — "kitne terms" wale problems solve karne ke liye use hota hai.
Gauss Summation Trick — $S_{n}$ ke peeche pairing idea.
Linear Functions — $a_{n}$ , $n$ mein ek straight line hai.

AP ko kya define karta hai?

Consecutive terms ka difference constant hota hai,

a_{n + 1} - a_{n} = d

AP ke nth term ka formula?

a_{n} = a + (n - 1) d

(n - 1) d

kyun,

n d

kyun nahi?

Pehli term ke liye 0 jumps chahiye; term

n

tak pahunchne mein sirf

n - 1

jumps of size

d

lagte hain.

n terms ka sum (standard form)?

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

n terms ka sum (average/first-last form)?

S_{n} = \frac{n}{2} (a + a_{n})

, yani terms × first aur last ka average.

Partial sums se

a_{n}

kaise nikalte hain?

a_{n} = S_{n} - S_{n - 1}

Agar

a_{n}

n

mein linear hai, toh yeh kya batata hai?

Sequence ek AP hai (uska common difference slope hai).

Agar

S_{n}

n

mein quadratic hai (bina constant term ke), toh sequence kya hai?

Ek AP.

2, 5, 8, 11

ka common difference?

d = 3

(consecutive terms subtract karo).

S_{n}

derive karne ka trick?

Sum ko aage + ulta likhо aur add karo; har column

a + a_{n}

equal hota hai.