IHATESAT

Hey. The following are questions that I got from an old worksheet on sequences, but I can't find the answers....

I thought that I would post them here to give you guys some practice, but also since I want to see if I am correct.

REMEMBER: THESE QUESTIONS ARE EXTREMELY HARD, BUT SHOULD MATCH THE DIFFICULTY OF THOSE ON THE EXAM.

Find the next number and GENERAL TERM for each of the following sequence. Justify each answer.

1) 3,8,15,24,35,48...

2) 6,14,18,28,30,42...

3) 1/6, 1/3, 1/2, 2/3, 5/6, 1,...

4) 4,1,0,1,4,9...

5) 2,8,26,80,242,728...

6) 15,210,63,220,243,230...

7) 2,7,4,14,6,21...

8) -1,4,1,6,3,8

9) 1,5,13,29,61,125...

10) 10,21,33,46,60,75...

Thanks guys. Good luck! (I doubt any of you guys can figure out the answers anyway ).

lil_killer129

is 10) 91?

lil_killer129

is 9) 253?

IHATESAT

I need the general formula... thanks!

mattd1688

3,8,15,24,35,48.
5 7 9 11 13 15
2 2 2 2 2

so 15+48 = 63.

This can be done for all of them to find the next term. As far as the pattern, um no clue. This was on the WPI programming contest and I didnt have a clue.


4) 4,1,0,1,4,9...
(n-1)^2 assuming that n starts at 1 and its 16


2) 6,14,18,28,30,42...
for the odds its 6n, and for the even its 7n
the seventh term is 6*7 = 42.

7) 2,7,4,14,6,21...
odd = 2n
even = 7n

8) -1,4,1,6,3,8
odd = n-2
even = 2n

3) 1/6, 1/3, 1/2, 2/3, 5/6, 1,...
it goes 1*6 * n(n-1) but I cant make this a formula.

stanfordream

#10
10n+(n-1)(n)/2
Solution: first notice that its a sequence of 0 1 3 6 10... plus 10n. well, 0 1 3 6 10... is known as triangle number and its explicit formula is (n-1)n/2. i don't know how to derive this explicit formula, but that's what it is and i memorized it. Thus, the answer is 10n+(n-1)(n)/2

stanfordream

#3 is arithmetic. it's just n/6

surge

These are ridiculously easy and by no means need calculus to solve. If you want I can post solutions... just tell me the numbers you want.

I dunno about the AP Calculus test, but you will probably NOT encounter these types of problems... if there was an AP Algebra2 test then you definitely would.

IHATESAT

Surge... what are the answers for #1-10 then... (big shot)..

haha.

Thanks

IHATESAT

Could you post the forumlas for #1-10, if they are "so-easy.""

I would really appriciate it.

Thanks

surge

Sorry, if I came off as a "big shot"...

lemme just finish my spanish homework and I'll post 'em.

IHATESAT

Thank you surge... I really appriciate it!

Evilbooyaa

mattd's answers - i think he assumed the first value is for n = 1, I would change the n's to (n+1)(all my edits are going off his technique)
#4 doesn't seem to work..... at n=1 in that, the answer would be 0.

#2 looks good
#7 - almost right. for odds its not 2n, its 1+ n
#8 looks good

stanfordream
#10 doesn't work for me..... if you plug in 1 for n, the result isn't 1., its 10.

surge
#3 is exactly right. nice spot.

mine
(btw, this is assuming that the first value is at n = 0....... all of my analyses.)
#1 look at pattern for odds
3 = 1 x 3
15 = 3 x 5
35 = 5 x 7
next number = 63 = 7 x 9

for evens, its
8 = 2 x 4
24 = 4 x 6
48 = 6 x 8

so general equation = (n+1)(n+3)

#4 (n-2)^2 -next number is 16.
#5 3 times the previous number + 2 (not sure how to write that in symbols)- next number = 2186
#6 odds- 3*3^(n-1)*(5+n)
evens- 200 + 10n
(for some reason, I can't get #6 to work out w/ the first value correlating to n = 0..... it has to be n= 1.

#10 previous number + 10 + n -next number is 91


there. skipped 9 cause i couldn't do it quick.
that should be all of them.

surge

All answers I believe have been posted except for 9. Number 9 is a special case... it cannot be solved using finite differences. Probably because it is an exponential equation.

HINT: For all other problems
Use finite differences.
Set up coefficient matrices in your calculator for linear, quadratic, cubic and quartic.
Take the inverse of your coefficient matrix (after you figure out what power its to by using finite differences) and multiply it by your solution matrix (which should nx1, where n is one higher than the degree of your polynomial and equal to the number of rows in your coefficient matrix).

EDIT: woops, some people posted recursive formulas (regarding the last term). Lemme post some explicit formulas.

surge

#10 explicit= (1/2)x[x+19]