How Do You Derive the General Formula for Given Recursive Sequences?

[lohengrin]

Hi guys, I'm doing some exercises in which given a recursive sequence and its first term, I have to find the general formula/term. I am stuck in two and I would like some help. Thanks in advance. Now, the sequences:


1) a1=1, an+1= an + ((-1)^(n+1))n^2

So, the first terms are: a2=2, a3=-2, a4=7, a5=-9, a6=16

Maybe you find it easy, I've been trying really hard to come up with the formula but I haven't got a clue.

2) a1=1, an+1= an + n^3

So, the first ones are: a2=2, a3=10, a4=37, a5=101

And again, no idea.

I suck at this, sorry to bother you but I would really appreciate your help.

 

[Ray Vickson]

Hi guys, I'm doing some exercises in which given a recursive sequence and its first term, I have to find the general formula/term. I am stuck in two and I would like some help. Thanks in advance. Now, the sequences:


1) a1=1, an+1= an + ((-1)^(n+1))n^2

So, the first terms are: a2=2, a3=-2, a4=7, a5=-9, a6=16

Maybe you find it easy, I've been trying really hard to come up with the formula but I haven't got a clue.

2) a1=1, an+1= an + n^3

So, the first ones are: a2=2, a3=10, a4=37, a5=101

And again, no idea.

I suck at this, sorry to bother you but I would really appreciate your help.

Use brackets. What you *wrote* says $a_n + 1 = a_n + (-1)^{n+1} n^2,$ but I am guessing meant $a_{n+1} = a_n + (-1)^{n+1} n^2.$ To do this in plain text, just say a_{n+1} or a_(n+1) instead of an+1, etc. Alternatively, you can use the "X₂" button on the menu bar above the input panel; that would give a_n+1.

Hint for 1): look at d_n = a_{n+1} - a_n = (-1)^(n+1) n^2 and note that
a_n - a_1 = d_{n-1} + d_{n-2} + ... + d_1.

RGV