If each term of a geometric progression a1, a2, a3 ……. with and a1 ≠ a2, is the arithmetic mean of the next two terms and Sn = a1 + a2 +……..+ an, then S20

Engineering

Mathematics

Geometric Progression and Its Properties

Question

If each term of a geometric progression a₁, a₂, a₃ ……. with $a_{1} = \frac{1}{8}$ and a₁ ≠ a₂, is the arithmetic mean of the next two terms and S_n = a₁ + a₂ +……..+ a_n, then S₂₀ – S₁₈ is equal to

2¹⁸

– 2¹⁸

2¹⁵

– 2¹⁵

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$T_{n} = \frac{T_{n + 1} + T_{n + 2}}{2}$

⇒ 2ar^n–1 =arⁿ + arⁿ⁺¹
⇒ r² + r = 2 ⇒ r = 1 ro – 2
⇒ r ≠ 1 ∵ (a₁ ≠ a₂) ⇒ r = – 2
Now
S₂₀ – S₁₈ = T₁₉ + T₂₀
= ar₁₈ + ar₁₉
= ar¹⁸(1 + r)

$\frac{1}{8} (- 2)^{18} (- 1) = - 2^{15}$