series

[stable vault]

just found this in one of my book back then and encountered one of the questions i still havent gotten it. Finding the general formula of (un) based on u1 and un+1, please help me with the 2 questions here

[rugged ledge]

for solving a complex recurrence relation, sometimes the best approach is to calculate the first few terms and look for a pattern.

in (3), u_1~u_4 are as follows

$u_1=1 \ u_2=-7/4 \ u_3=-32/11 \ u_4=-99/25$

[solid crown]

へいほー

[rugged ledge]

from this, we can presume that u_n will be like -n + (small value)

so now, let u_n = v_n -n

and ...

In (4), the relation is not as complex as the one in (3).

and when we decompose frac in right side into partial fracs,

we can easily presume that the original relation will be like

$u_{n+1}-\frac{A}{n+2}=\frac{2}{3}\left(u_n -\frac{A}{n+1}\right)$

[solid crown]

へいほー

[rugged ledge]

and ...