Why does ∑ k = 1 n 1 4 ( 1 4 k − 3...

Extrakt04

Extrakt04

Answered

2022-06-28

Why does k = 1 n 1 4 ( 1 4 k 3 1 4 k + 1 ) turn into 1 4 ( 1 1 4 n + 1 )

Answer & Explanation

hofyonlines5

hofyonlines5

Expert

2022-06-29Added 12 answers

k = 1 n ( 1 4 k 3 1 4 k + 1 )
is a "telescoping series". Write it out explicitly, and note most of the terms cancel.
k = 1 n ( 1 4 k 3 1 4 k + 1 ) = ( 1 1 1 5 ) + ( 1 5 1 9 ) + ( 1 9 1 13 ) + + ( 1 4 n 7 1 4 n 3 ) + ( 1 4 n 3 1 4 n + 1 ) = 1 1 4 n + 1 .
Cory Patrick

Cory Patrick

Expert

2022-06-30Added 6 answers

k = 1 n ( 1 4 k 3 1 4 k + 1 )
= ( 1 1 1 5 ) + ( 1 5 1 9 ) + + ( 1 4 n 7 1 4 n 3 ) + ( 1 4 n 3 1 4 n + 1 )
= 1 1 ( 1 5 1 5 ) ( 1 9 1 9 ) ( 1 4 n 3 1 4 n 3 ) 1 4 n + 1
= 1 1 4 n + 1
The (*) line should be 1 4 ( 1 1 4 n + 1 ), not 1 4 ( 1 1 4 k + 1 )

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