If t 4 = 4 , t 5 = 5 and t n = 2 t n − 1 − t n − 2 for n ≥ 3 , find ∑ n = 1 2009 t n .

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MathematicsSequence & SeriesA.P.Medium2 minQB

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If $t_{4} = 4, t_{5} = 5$ and $t_{n} = 2 t_{n - 1} - t_{n - 2}$ for $n \geq 3$ , find $\sum_{n = 1}^{2009} t_{n}$ .

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Answer:

Solution:

The recurrence makes successive differences constant. As $t_{5} - t_{4} = 1$ , backward induction gives $t_{n} = n$ ; hence the sum is $2009 \cdot 2010/2 = 2019045$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:A.P.

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