Let f ( 1 ) = 1 and f ( n ) = 2 ∑ r = 1 n − 1 f ( r ) , then ∑ n = 1 m f ( n ) is equal to

Question

Let f ( 1 ) = 1 and f ( n ) = 2 ∑ r = 1 n − 1 ​ f ( r ) , then ∑ n = 1 m ​ f ( n ) is equal to

TestHub · Accepted Answer

f ( n ) = 2 ( f ( 1 ) + f ( 2 ) + … + f ( n − 1 )) ∴ f ( n + 1 ) = 2 ( f ( 1 ) + f ( 2 ) + … . + f ( n )) ⇒ f ( n + 1 ) = 3 f ( n ) for n ≥ 2 . Also f ( 2 ) = 2 f ( 1 ) = 2 , f ( 3 ) = 3 f ( 2 ) = 2 ⋅ 3 . ∑ n = 1 m ​ f ( n ) = f ( 1 ) + f ( 2 ) + … + f ( m ) = 1 + 2 + 2.3 + 2. 3 2 + … + 2. 3 m − 2 = 1 + 2 ( 1 + 3 + 3 2 + … + 3 m − 2 ) = 3 m − 1

Mathematics - Sequence & Series Question with Solution | TestHub

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