1. ## Recurrence/APGP/Sigma Notation

1) Un is defined by recurrence relation:
u1=2 and u(n)=u(n-1)+1-1/(n(n-1)) for n>=2

By using method of diff, find u(n) (Ans: (n^2)/(n+1))

Some quick help wud be great! thanks

2. Hello, zeromeyzl!

Could you restate the problem?
As given, it doesn't compute . . .

$\displaystyle u(n)$ is defined by recurrence relation:
. . $\displaystyle u(1)\:=\:2,\;\;u(n)\:=\:u(n-1)+1-\frac{1}{n(n-1)}\;\;\text{ for }n \geq 2$

By using method of diff, find $\displaystyle u(n)$

Answer: .$\displaystyle \frac{n^2}{n+1}$ . . . . not correct

If I read everything correctly, we have:

. . $\displaystyle u(1) \;=\;2$

. . $\displaystyle u(2) \;=\;2 + 1 - \frac{1}{2\cdot1} \;=\;3 - \frac{1}{2} \;=\;\frac{5}{2}$

. . $\displaystyle u(3) \;=\;\frac{5}{2} + 1 - \frac{1}{3\cdot2} \;=\;\frac{7}{2} - \frac{1}{6} \;=\;\frac{20}{6} \;=\;\frac{10}{3}$

. . $\displaystyle u(4) \;=\;\frac{10}{3} + 1 - \frac{1}{4\cdot3} \;=\;\frac{13}{3} - \frac{1}{12} \;=\;\frac{51}{12} \;=\;\frac{17}{4}$

The general term for this sequence is: .$\displaystyle u(n) \;=\;\frac{n^2+1}{n}$