Year 12s: what will be the first way you research your future options? >>

Recursive definition

Write down the first five terms of the sequence and give the recursive definition.
I'm not sure how to find the recursive definition.

ur=r^2
I worked the first 5 terms 1,4,9,16,25

ur=1/2r(r+1)
First 5 terms 1,3,6,10,15

ur=1/6r(r+1)(2r+1)
First 5 terms 1,5,14,30,55

Reply 1

Zacken

Original post by osayukiigbinoba

Write down the first five terms of the sequence and give the recursive definition.
I'm not sure how to find the recursive definition.

ur=r^2
I worked the first 5 terms 1,4,9,16,25

Can you see that if you take the previous number and add a odd number to it you get the next term in the sequence?

So, for example
1 + 3 = 4
4 + 5 = 9
9 + 7 = 16
16 + 9 = 25

Reply 2

yuki11

Original post by Zacken

Can you see that if you take the previous number and add a odd number to it you get the next term in the sequence?

So, for example
1 + 3 = 4
4 + 5 = 9
9 + 7 = 16
16 + 9 = 25

Yeah that makes sense thank you. My book says the answer is ur+1=ur+2r+1
So does the 2r+1 represent the odd sequence you mentioned?

Reply 3

Zacken

Original post by osayukiigbinoba

Yeah that makes sense thank you. My book says the answer is ur+1=ur+2r+1
So does the 2r+1 represent the odd sequence you mentioned?

Yep. Try it out for

r=1, 2, 3, 4, \cdots

and see if it generates the sequence you want.

Reply 4

yuki11

Original post by Zacken

Yep. Try it out for

r=1, 2, 3, 4, \cdots

and see if it generates the sequence you want.

It works thank you. Should I do something similar for the other questions?

Reply 5

Zacken

Original post by osayukiigbinoba

It works thank you. Should I do something similar for the other questions?

Yep, try and see how.

Reply 6

yuki11

Original post by Zacken

Yep, try and see how.

For 1,3,6,10,15 I got ur+1= ur+r+1
But for 1,5,14,30,55 I have no idea. All I worked out so far was the difference 4,9,16,25 between each term. Then the second difference is 5,7,9.

Reply 7

Zacken

Original post by osayukiigbinoba

For 1,3,6,10,15 I got ur+1= ur+r+1

Looks good!

But for 1,5,14,30,55 I have no idea. All I worked out so far was the difference 4,9,16,25 between each term. Then the second difference is 5,7,9.

What does the bolded bit remind you of...? Look at the first question for a reminder.

Answer in spoiler:

Spoiler

(edited 7 years ago)

Reply 8

yuki11

Original post by Zacken

Looks good!

What does the bolded bit remind you of...? Look at the first question for a reminder.

Answer in spoiler:

Spoiler

Could you explain where (r+1)^2 comes from please?

Reply 9

Zacken

Original post by osayukiigbinoba

Could you explain where (r+1)^2 comes from please?

You have:

1 + 2^2 = 5
5 + 3^2 = 14
14 + 4^2 = ...

In all cases you have:

1st term: 1st term
2nd term: 1st term + 2^2
3rd term: 2nd term + 3^2
4th term: 3rd term + 4^2
(n+1)th term: (n)th term + (n+1)^2

Reply 10

yuki11