Let S k , k = 1 , 2 , … , 100 , denote the sum of the infinite geometric series whose first term is k ! k − 1 and the common ratio is k 1 . Then the value of 100 ! 10 0 2 + ∑ k = 1 100 ( k

Question

Let S k ​ , k = 1 , 2 , … , 100 , denote the sum of the infinite geometric series whose first term is k ! k − 1 ​ and the common ratio is k 1 ​ . Then the value of 100 ! 10 0 2 ​ + ∑ k = 1 100 ​ ​ ( k 2 − 3 k + 1 ) S k ​ ​ is

TestHub · Accepted Answer

We have, S k ​ = 1 − k 1 ​ k ! k − 1 ​ ​ = ( k − 1 )! 1 ​ Now, ​ ( k 2 − 3 k + 1 ) S k ​ = {( k − 2 ) ( k − 1 ) − 1 } = ( k − 3 )! 1 ​ − ( k − 1 )! 1 ​ ⇒ k = 1 ∑ 100 ​ ​ ( k 2 − 3 k + 1 ) S k ​ ​ = 1 + 1 + 2 − ( 99 ! 1 ​ + 98 ! 1 ​ ) = 4 − 100 ! 10 0 2 ​ ​ ⇒ 100 ! 10 0 2 ​ + k = 1 ∑ 100 ​ ​ ( k 2 − 3 k + 1 ) S k ​ ​ = 4

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