During the finals of a state-level math olympiad at PSG Public School, Coimbatore, students were given this challenge:
If α and β are the roots of the equation 15x² - 8x + 1 = 0, what is the value of the following:
(α + β) + (α² + β²) + (α³ + β³) + (α⁴ + β⁴) + ...?
What answer did Shreya give?
Note: Round off your answer to 3 decimal places. For example, if the answer is 0.5 put 0.500, if the answer is 0.5447 then put 0.545, and if the answer is 0.7825 then put 0.783.
Answer (2)
The value of the series ( α + β ) + ( α 2 + β 2 ) + ( α 3 + β 3 ) + … is approximately 1.143 when rounded to three decimal places.
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To solve this problem, we need to determine the sum given as:
( α + β ) + ( α 2 + β 2 ) + ( α 3 + β 3 ) + ( α 4 + β 4 ) + …
Given α and β are the roots of the quadratic equation:
15 x 2 − 8 x + 1 = 0
From Vieta's formulas, we know:
α + β = 15 − ( − 8 ) = 15 8
α β = 15 1
To find the sum α n + β n , we use the recurrence relation derived from the relation α n + β n = ( α + β ) ( α n − 1 + β n − 1 ) − α n − 2 β n − 2 ( α β ) , which simplifies using:
S n = ( α + β ) S n − 1 − ( α β ) S n − 2
where S n = α n + β n and where initial conditions are S 0 = 2 and S 1 = α + β .
This describes a geometric series:
First, calculate S 2 = ( α + β ) 2 − 2 α β = ( 15 8 ) 2 − 2 ⋅ 15 1
Find the common ratio ( 15 8 ) . Effectively, this series is a geometric one.
The first term of the series is S 1 = α + β = 15 8 .
Therefore, the infinite series sum is:
Sum = 1 − r S 1 = 1 − 15 8 15 8 = 15 7 15 8 = 7 8
Rounding 7 8 to three decimal places gives:
1.143
Therefore, Shreya's answer to the olympiad problem was approximately 1.143 .
