Sequence and series

Ques1) If one Geometric mean G and two arithmetic means p & q are inserted between two quantities, then show that
G² = (2p-q)(2q-p)

Ques2) Let a,b,c are respectively the sum of the first n terms, the next n terms, and the next n terms of a G.P. Show that a,b,c are in G.P.

1 Answers

Let A and B be te two numbers

G² = AB

now, A,p,q,B are in A.P

common difference = B-A/3

=> p= A + B-A/3 = 2A+B/3
and q = A + 2(B-A)/3 = 2B+A/3

now (2p-q)(2q-p)= AB putting values of p and q

thus AB= G² = (2p-q)(2q-p)

Hence proved

√ √ || {} [] () → ~ ^ x x x → → ln log lg lim lim cos sin tan cot arcsin arccos arctan

∫ ∫ ∫ ∫ ∫ ∬ ∬ ∬ ∬ ∬ ∭ ∭ ∭ ∭ ∭ ⨌ ⨌ ⨌ ⨌ ⨌

∑ ∑ ∑ ∑ ∑ ∮ ∮ ∮ ∮ ∮ ∏ ∏ ∏ ∏ ∏ ∐ ∐ ∐ ∐ ∐

+ - × ÷ ± ∓ = ∪ ∩ ≠ ∼ ≈ ∅ ∀ ∃ ∈ ∋ ∝ ≤ ≥ ≃ ≅ ≡ < > ≪ ≫ ⊂ ∉ ⊃ ⊆ ⊇ ⇒ ⇐ ⇔ ⇒ ⇐ ⇔ ⇑ ⇓ ⇕ → ← ↔ → ← ↔ ↑ ↓ ↕ ↖ ↗ ↘ ↙ ↪ ↩ ⇀ ↼ ⇁ ↽

° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω