Question

please give me the step wise solution of question 19

Answer 1

Heya User,

--> We know that :->
   --> ( x - 1 )( xⁿ + xⁿ⁻¹ + ... + x² + x + 1 ) = ( xⁿ⁺¹ - +x )

--> However, for x = 2
             --> ( 2 - 1 )( 2ⁿ + 2ⁿ⁻¹ + ... + 2² + 2 + 1 ) = ( 2ⁿ⁺¹ - 2 ) 
             => ( 2ⁿ + 2ⁿ⁻¹ + ... + 2² + 2 + 1 ) = ( 2ⁿ⁺¹ - 2 )

Now, considering the above series :->

--> T₁ + T₂ + T₃  + ... + T(n) = [ 2¹ + 2² + ... + 2ⁿ ] + 2[ 1 + 2 + ... + n]
                                           = [ 2ⁿ⁺¹ - 2 ] + n ( n + 1 ) 

Hence, Σ T(n) = [ 2ⁿ⁺¹ - 2 ] + n ( n + 1 )

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One line solution -->
Σ T(n) = Σ ( 2ⁿ + 2n ) = Σ ( 2ⁿ) + 2 Σn = [ 2ⁿ⁺¹ - 2 ] + n ( n + 1 )