How do you identify if a sequence is AP or GP?

AP: Subtract consecutive terms — if the difference (d = aₙ₊₁ − aₙ) is constant, it's an AP. GP: Divide consecutive terms — if the ratio (r = aₙ₊₁/aₙ) is constant, it's a GP. Example: 2, 5, 8, 11 → d=3 (AP). 3, 6, 12, 24 → r=2 (GP). 1, 2, 4, 8 → r=2 (GP).

What is the condition for an infinite GP to have a finite sum?

An infinite GP has a finite sum only when |r| < 1 (i.e., −1 < r < 1). In this case S∞ = a/(1−r). If |r| ≥ 1, the terms don't decrease and the series diverges to infinity. Example: 1/2 + 1/4 + 1/8 + ... has r = 1/2 < 1, so S∞ = (1/2)/(1−1/2) = 1.

What are the three assumptions when terms of an AP are taken conveniently?

For 3 terms in AP: take (a−d), a, (a+d) — sum = 3a, product more manageable. For 4 terms in AP: take (a−3d), (a−d), (a+d), (a+3d) — sum = 4a, common difference = 2d. For 5 terms: (a−2d), (a−d), a, (a+d), (a+2d) — sum = 5a.

Arithmetic Progression and Geometric Progression

Sequences and Series • Class 11 Mathematics • NCERT • CBSE

AP: nth term aₙ = a + (n−1)d; sum Sₙ = n/2[2a + (n−1)d]. GP: nth term aₙ = arⁿ⁻¹; sum Sₙ = a(rⁿ−1)/(r−1). Arithmetic Mean (AM) = (a+b)/2; Geometric Mean (GM) = √(ab). AM ≥ GM for positive numbers.

Key Formulas

aₙ = a + (n−1)d (AP nth term)
Sₙ = n/2[2a + (n−1)d] (AP sum)
aₙ = arⁿ⁻¹ (GP nth term)
S∞ = a/(1−r) for |r|<1 (infinite GP)
AM ≥ GM: (a+b)/2 ≥ √(ab)

Frequently Asked Questions

How do you identify if a sequence is AP or GP?: AP: Subtract consecutive terms — if the difference (d = aₙ₊₁ − aₙ) is constant, it's an AP. GP: Divide consecutive terms — if the ratio (r = aₙ₊₁/aₙ) is constant, it's a GP. Example: 2, 5, 8, 11 → d=3 (AP). 3, 6, 12, 24 → r=2 (GP). 1, 2, 4, 8 → r=2 (GP).
What is the condition for an infinite GP to have a finite sum?: An infinite GP has a finite sum only when |r| < 1 (i.e., −1 < r < 1). In this case S∞ = a/(1−r). If |r| ≥ 1, the terms don't decrease and the series diverges to infinity. Example: 1/2 + 1/4 + 1/8 + ... has r = 1/2 < 1, so S∞ = (1/2)/(1−1/2) = 1.
What are the three assumptions when terms of an AP are taken conveniently?: For 3 terms in AP: take (a−d), a, (a+d) — sum = 3a, product more manageable. For 4 terms in AP: take (a−3d), (a−d), (a+d), (a+3d) — sum = 4a, common difference = 2d. For 5 terms: (a−2d), (a−d), a, (a+d), (a+2d) — sum = 5a.

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