Difference method for some finite algebraic series

A good seriesYue Kwok Choy

:::

ProofWe can prove the (k formula by :

(1)Mathematical Induction proof omitted.

(2)Difference method

The general term =

Let

. Then

(

Why the series (k formula is good ?(1)(1, (2 ,, (k formulas show certain pattern and are easy to remember .

(2)The formulas can be used to find some other finite series .

Example 1

Find

(1)

and (2)

(1)

(2)

(

Example 2

Find the sum of the first n terms :

We wish to find .

Consider (r + 1) (2r + 3) ( A r(r + 1) + B r + C

We can use compare coefficient method, or otherwise, and find that : A = 2 , B = 3 , C = 3((r + 1) (2r + 3) ( 2 r(r + 1) + 3 r + 3

(

Exercise

Show that :

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