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[Solved] Sequences

 
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June 15, 2021 10:53 am
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Staff 
(@armor)
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If sequence \left \{ a_{n} \right \}  satisfies a_{1}=3,   and a_{n+1}=2a_{n}+1(n\epsilon \mathbb{N}).  Find the general term of the sequence.

 

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June 15, 2021 10:53 am
Topic starter
Staff 
(@armor)
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Adding 1 to both sides of the equation \small a_{n+1}=2a_{n}+1  to obtain \small a_{n+1}+1=2(a_{n}+1).  We apply the recurrent relation to obtain \small a_{n+1}+1=2(a_{n}+1),     \small a_{n-1}+1=2(a_{n-2}+1),........., a_{2}+1=2(a_{1}+1). 

Multiplying the above equations and applying \small a_{1}=3  to generate \small a_{n}=2^{n+1}-1 (n\epsilon \mathbb{N}).

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