Problem 116 Solution

a₀ = 0

a₁ = 9/19*a₂
a₂ = 9/19*a₃ + 10/19*a₁
a₃ = 9/19*a₄ + 10/19*a₂
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a₂₇ = 9/19*a₂₈ + 10/19*a₂₆
a₂₈ = 9/19*a₂₉ + 10/19*a₂₇
a₂₉ = 9/19*a₃₀ + 10/19*a₂₈
a₃₀ = 1

Divide the left side into two parts:

9/19*a₁ + 10/19*a₁ = 9/19*a₂
9/19*a₂ + 10/19*a₂ = 9/19*a₃ + 10/19*a₁
9/19*a₃ + 10/19*a₃ = 9/19*a₄ + 10/19*a₂
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9/19*a₂₇ + 10/19*a₂₇ = 9/19*a₂₈ + 10/19*a₂₆
9/19*a₂₈ + 10/19*a₂₈ = 9/19*a₂₉ + 10/19*a₂₇
9/19*a₂₉ + 10/19*a₂₉ = 9/19*a₃₀ + 10/19*a₂₈

Rearange with 10/19 terms on the left side and 9/19 terms on the right:

10/19*(a₁) = 9/19*(a₂ - a₁)
10/19*(a₂ - a₁) = 9/19*(a₃ - a₂)
10/19*(a₃ - a₂) = 9/19*(a₄ - a₃)
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10/19*(a₂₇ - a₂₆) = 9/19*(a₂₈ - a₂₇)
10/19*(a₂₈ - a₂₇) = 9/19*(a₂₉ - a₂₈)

Next multiply both sides by 19/9:

10/9*(a₁) = (a₂ - a₁)
10/9*(a₂ - a₁) = (a₃ - a₂)
10/9*(a₃ - a₂) = (a₄ - a₃)
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10/9*(a₂₇ - a₂₆) = (a₂₈ - a₂₇)
10/9*(a₂₈ - a₂₇) = (a₂₉ - a₂₈)

Next telescope sums:

(a₂ - a₁) = 10/9*(a₁)
(a₃ - a₂) = (10/9)²*(a₁)
(a₄ - a₃) = (10/9)³*(a₁)
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(a₂₉ - a₂₈) = (10/9)²⁸*(a₁)
(a₃₀ - a₂₉) = (10/9)²⁹*(a₁)

Next add the above equations:

(a₃₀ - a₁) = a₁ * ((10/9) + (10/9)² + (10/9)³ + ... + (10/9)²⁹)

1 = a₁ * (1 + (10/9) + (10/9)² + (10/9)³ + ... + (10/9)²⁹)

a₁ = 1 / (1 + (10/9) + (10/9)² + (10/9)³ + ... + (10/9)²⁹)

a₁ = (10/9 - 1) / ((10/9)³⁰ - 1)

Now that we know a₁ we can find a₂₀:

(a₂ - a₁) = 10/9*(a₁)
(a₃ - a₂) = (10/9)²*(a₁)
(a₄ - a₃) = (10/9)³*(a₁)
.
.
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(a₁₉ - a₁₈) = (10/9)¹⁸*(a₁)
(a₂₀ - a₁₉) = (10/9)¹⁹*(a₁)

Add the above equations together:

(a₂₀ - a₁) = a₁ * ((10/9) + (10/9)² + (10/9)³ + ... + (10/9)¹⁹)
a₂₀ = a₁ * ((10/9)²⁰ - 1)) / (10/9 - 1))
a₂₀ = [ (10/9 - 1) / ((10/9)³⁰ - 1) ] * [ ((10/9)²⁰ - 1) / (10/9 - 1) ]
a₂₀ = ((10/9)²⁰ - 1) / ((10/9)³⁰ - 1) =~ .3198