Not Limiting a Sum
α , β are the roots of 3 7 5 x 2 − 2 5 x − 2 = 0 . Denote S n = α n + β n , and if the summation below is in the form of b a where a , b are coprime positive integers. Find the value of a + b .
n = 1 ∑ ∞ S n
The answer is 13.
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2 solutions
Please proof that G P approaches to infinity are eligible!
Shit man I didn't cancel the terms ...:(
x 2 − 1 5 1 x − 3 7 5 2 = 0
Newton's Sum:
S 1 − 1 5 1 S 2 − 1 5 1 S 3 − 1 5 1 ⋮ S 1 − 3 7 5 4 S 2 − 3 7 5 2 S 1 = 0 = 0 = 0
Sum all the equations together we get
( ∑ S n ) − 1 5 1 ( ∑ S n ) − 1 5 1 − 3 7 5 2 ( ∑ S n ) − 3 7 5 4 ∑ S n = 0 = 1 2 1
There're some typos in your 3rd Newton sum and final equation
= n → ∞ lim α + β + α 2 + β 2 + α 3 + β 3 . . . . α n + β n
collect α a n d β t e r m s applying infinite GP formula
we get 1 − α α + 1 − β β
further simplification gives 1 − ( α + β ) − α β α + β − 2 α β
substitute the value α + β = 3 7 5 2 5 a n d α β = 3 7 5 − 2
u will get 3 4 8 2 9 = 1 2 1