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Answer:
75025 / 13201 ≈ 5.68328
Step-by-step explanation:
As α is a zero of x² - 5x - 5, it satisfies α² = 5α + 5.
Multiplying this by αⁿ gives αⁿ⁺² = 5αⁿ⁺¹ + 5αⁿ.
Similarly, we have βⁿ⁺² = 5βⁿ⁺¹ + 5βⁿ.
Taking the difference between these equations now gives
aₙ₊₂ = 5aₙ₊₁ + 5aₙ. (*)
We just need to initial values to work out later ones.
Easiest, we have a₀ = α⁰ - β⁰ = 1 - 1 = 0.
Next, to get a₁, notice that
a₁² = ( α - β )² = ( α + β )² - 4αβ = 5² - 4(-5) = 25 + 20 = 45
where the values for α+β and αβ come from the coefficients of the quadratic.
Thus a₁ = ±√45. Since it's not specified (and it ultimately won't matter) which root is α and which is β, we may choose α to be the larger of the two so that a₁ is positive, making a₁ = √45.
With these two starting values, we can use the relation (*) to calculate the values up to a₁₂ :
0, 6.708203932499369, 33.54101966249685, 201.24611797498113, 1173.93568818739, 6875.909030811856, 40249.22359499623, 235625.6631290404, 1379374.433620183, 8075000.483746117, 47271874.586831495, 276734375.35288805, 1620031249.698598
The required answer is then (approximately)
( 1620031249.698598 - 47271874.586831495 ) / 276734375.35288805
≈ 5.68328.
To get the exact answer 75025 / 13201, just start by making the definition
bₙ = aₙ / ( α - β ).
Then ( a₁₂ - a₁₀ ) / a₁₁ = ( b₁₂ - b₁₀ ) / b₁₁, so the process is the same, but the values that we calculate are all integers since b₀ = 0 and b₁ = 1.
The sequence of values up to b₁₂ are:
0, 1, 5, 30, 175, 1025, 6000, 35125, 205625, 1203750, 7046875, 41253125, 241500000
so the required answer is (exactly!)
(241500000 - 7046875) / 41253125 = 75025 / 13201
Hope this helps!