Quadratic in Recurrence relation

Hi, first time poster.

**Q.** In the sequence u1, u2, u3,…,un,

u1 = 0, u2 = 3, u3 = 12 and un = a + bn + cn2

Find the values of a, b and c.

**Attempt: **

If n = 1 then u1 = a + b(1) + c(1)2 = 0

= a + b + c = 0

If n = 2 then u2 = a + b(2) + c(2)2 = 3

= a + 2b + 4c = 3

If n = 3 then u3 = a + b(3) + c(3)2 = 12

= a + 3b + 9c = 12

= a + b + c = 0

= a + 2b + 4c = 3

= a + 3b + 9c = 12

It's been a number of years since I've last studied maths. Can anyone help me on what comes next? Many thanks.

Re: Quadratic in Recurrence relation

Quote:

a + b + c = 0

a + 2b + 4c = 3

a + 3b + 9c = 12

subtract 1st equation from the 2nd ...

b + 3c = 3

subtract 2nd equation from the 3rd ...

b + 5c = 9

solve for b and c using these two equations

Re: Quadratic in Recurrence relation

Are you able to use the linear combination method to solve systems?

There's also one mistake, the last equation has to be: $\displaystyle a+3b+6c=12$

Re: Quadratic in Recurrence relation

Quote:

Originally Posted by

**GrigOrig99** Hi, first time poster.

**Q.** In the sequence u1, u2, u3,…,un,

u1 = 0, u2 = 3, u3 = 12 and un = a + bn + cn^2

Find the values of a, b and c.

**Attempt: **

If n = 1 then u1 = a + b(1) + c(1)2 = 0

= a + b + c = 0

If n = 2 then u2 = a + b(2) + c(2)2 = 3

= a + 2b + 4c = 3

If n = 3 then u3 = a + b(3) + c(3)2 = 12

= a + 3b + 9c = 12

= a + b + c = 0

= a + 2b + 4c = 3

= a + 3b + 9c = 12

It's been a number of years since I've last studied maths. Can anyone help me on what comes next? Many thanks.

a+b+c=0 gives a+2b+4c=(a+b+c)+b+3c=3=0+3

a+2b+4c=3 gives a+3b+9c=(a+2b+4c)+b+5c=12=3+9

Then b+3c=3 gives b+5c=(b+3c)+2c=9=3+6

Re: Quadratic in Recurrence relation

Quote:

Originally Posted by

**Siron** Are you able to use the linear combination method to solve systems?

There's also one mistake, the last equation has to be: $\displaystyle a+3b+6c=12$

the OP meant $\displaystyle a + bn + cn^2$ ... the original system is correct.

Re: Quadratic in Recurrence relation

Fantastic! I've just managed to work it out. Thanks for all the help guys. Awesome site.