## anonymous 5 years ago I need help with a sequence

1. anonymous

10 13 16 8 If i is multiple of 4 divide it by 2 if it is not a multiple of 4 add 3 Find the 500th term The 500th term is 8 but i need a formula to show how i worked it out

2. watchmath

10,13,16,8,11,14,17,20,10,13,16,8 Then we can see that $$a_4=a_{12}=8$$ and in general $$8=a_4=a_{4+8n}$$ for any $$n$$. Since $$500=4+8n\Leftrightarrow 496=8n\Leftrightarrow n=124$$ then $$a_{500}=8$$.

3. anonymous

1. 8+3=11 2. 11+3=14 3. 14+3=17 4. 17+3=20 5. 20/2=10 6. 13 7. 16 8. 8 this will be repeating now the 4th element is 8 plus every 4+k8 element (k is an integer) 500=4+62*8 so it is 8

4. watchmath

Andras is right for the $$n$$ :).

5. anonymous

The rule is if it is a multiple of 4 divide it by 2 so 8/2 =4/2=2+3=5+3=8/2=4/2=2+3=5

6. anonymous

that is what keeps on happening

7. anonymous

haha watchmath we both suck :)

8. anonymous

10 13 16 8 below four will be repeating than 4 2 5 8 so we can see that every 4th element is 8. 500/4=125 this proves that it is 8

9. anonymous

but 500 also divides by 4 so how can u be certain that it is 8

10. anonymous

every 4th element is an 8 (we can see that now) so if an element can be written as the the factor of 4 than it is 8. 1*4th element=8 2*4th =8 3*4th =8 and so on so the 125*4th element (500)=8

11. anonymous

ok thank you very much for helping me

12. anonymous

no prob

13. anonymous

What is the formula

14. anonymous

I dont think there is a formula here, you have to use your brain

15. anonymous

Are you sure because my dad says there is a formula but will not tell me

16. anonymous

the formula us that if the number can be written as 4k (k is an integer) than it is 8, I dont see what your dad wants.

17. anonymous

ok thanks i will go now and prove my dad wrong

18. anonymous

Good luck for that! :-)

19. watchmath

Well you can make a formula $$a_1=10,a_2=13,a_3=16$$ $$a_{4n}=8$$ for all $$n$$ $$a_{4n+1}=4$$ for $$n\geq 1$$ $$a_{4n+2}=2$$ for $$n\geq 1$$ $$a_{4n+3}=5$$ for $$n\geq 1$$

20. anonymous

can you make the formula for me pretty please

21. watchmath

I wrote the formula above! You need to free your mind! the formula doesn't always means that you have to write it as a single expression. The description above is count as formula too.

22. anonymous

ok i need to tell you something i am an 11 year old child in harrow that goes to Glebe Primary School my teacher gave me a sequnce and i had to find out the 500th term usin a simple formula do u think i can understand wat u just wrote

23. anonymous

at 11 you learn about sequences?? tell your teacher to relax a bit :-)

24. anonymous

i have to find a formula to be in this level six study group at school

25. watchmath

Ok, you must be smart since your teacher has a high hope for your and give you this problem. The formula that I gave above basically says that to find the nth terms of the sequence (were $$n\geq 4$$ )) we do the following: When n/4 has no remainder, then the nth term is 8 when n/4 has 1 as the remainder, then the nth term is 4 when n/4 has 2 as the remainder, then the nth term is 2 when n/4 has 3 as the remainder, then then nth term is 5 For example what is the 6666th term? Lets divide 6666/4. We have 1666 with remainder 2. So the 6666th term is 2.

26. anonymous

ooooooooooooo i get it

27. anonymous

so 500/4 is 125 with no remainder so it is 8

28. watchmath

perfect!!

29. anonymous

thank u very much u are the best u should get a job as a teacher

30. anonymous

wat do u mean by remainder 6666/4=1666.5 so how is it 2

31. watchmath

$$6666=4\cdot 1666 + 2$$ So the remainder is 2. You need to divide by hand to get the remainder.

32. anonymous

what is the hand

33. watchmath

By hand means not using calculator. No body teach to divide by "hand" anymore? :D

34. anonymous

how does 6666= 4.1666+2=

35. anonymous

i hav no idea where u get the remainder 2 from

36. watchmath

Do you know how to divide just using paper and pencil?

37. anonymous

yes i do

38. watchmath

Ok, try to divide 6666 by 4 using paper and pencil.

39. anonymous

ooooooo i see it now thank u very much u are the best

40. watchmath

good for you :D

41. anonymous

bye thank u so much