Question

how can we get next number in this series 4, 21, 143, 1061, 8363 ......

Answer 1

i can make you a formula that will work if you want <.<

It may not be what the creator of the problem intended though.

Answer 2

kk, pls do help me

Answer 3

alright, one sec.

Answer 4

and ya also teach me to make those kind of formulas so that next time i can do any such kind of problem

Answer 5

http://www.wolframalpha.com/input/?i=-1927%2F60%28-2%29^n%2B533%2F3%28-1%29^n%2B1364%2F3%281%29^n-3173%2F12%282%29^n%2B958%2F15%283%29^n%2C+n%3D5

Answer 6

you can change the value of n at the end of the function to see that it works, make n = 6 to see the 6th term.

The method for creating these involves a bit of linear algebra, if you arent very comfortable with Linear Algebra it might be a little complicated.

Answer 7

yeah show me too, although that has to be way uglier than intendend

Answer 8

good joe math , keep it up

Answer 9

i am good at algebra , teach me how to made that function

Answer 10

joe please also tell me about some software that i can use to plot graphs and functions and to get functions from plotted graphs ?

Answer 11

Alright. The idea is two-fold.

1st, we are going to abuse the fact that the tail of the sequence is unknown. We dont know if this sequence converges, diverges, whatever.

2nd, since we dont know about the part of the sequence beyond the 5th term, wouldnt it be nice if the 6th term was some linear combination of the first 5 terms? Who is to say it is or isnt?

Answer 12

ok , what next?

Answer 13

So, (without going into too much detail, since this might be my undergraduate thesis <.<),

Step 1, pick 5 integers that are all different. I picked -2, -1, 1, 2, 3.

Step 2 calculate their powers and make the, into a square matrix.
You end up with:

-2 -1 1 2 3
4 1 1 4 9
-8 -1 1 8 27
16 1 1 16 81
-32 -1 1 32 243

Lets call this matrix A

Answer 14

i don't mean to interrupt so i will let you continue, but i can tell you what this sequence represents if you like

Answer 15

Step 3: Solve this matrix equation for x:
\[Ax = b\]where b = (4, 21, 143, 1061, 8363)

Answer 16

the vector x will be the coefficients of the powers of the 5 numbers you chose.

Answer 17

y we did that ?

Answer 18

so after i chose the numbers -2, -1, 1, 2, 3, i knew the solution was going to be in the form:
\[c_1(-2)^n+c_2(-1)^n+c_3(1)^n+c_4(2)^n+c_5(3)^n\]
Solving that matrix equation tells me what the c's are.

Answer 19

hey, this kind of thing was not taught to us ?? it is given in some book or you made it ?

Answer 20

can you teach me applying that on some easy series ??? please

Answer 21

4, 21, 143, 1061, 8363, 68906, 586081, 5096876,

Answer 22

It wasnt something taught in a class per se...its kinda like working linear recurrences backwards...i did get the idea on my own. I doubt im the first person to think about it though, its not a super complicated idea.

Answer 23

saifoo it is your bed time!

Answer 24

@joe you know what these numbers are?

Answer 25

not at all! is there some pattern? now that i have this way of creating sequence formula's, ive given up on recognizing patterns lol

Answer 26

hey , u r bad? u scared me

Answer 27

@joe hey, please teach me by applying this on some easy series , like some arithmetic series please

Answer 28

if you had one you are famous. and how divanshu is supposed to come up with the next one is beyond me.
an is the number of primes of n digits.
1 digit, 2 primes,
2 digits, 21 primes,
3 digits, 143 primes

which makes me suspect that the question is a joke

Answer 29

should be
1 digit, 4 primes

Answer 30

LOL,

Answer 31

oh woah <.< thats ridiculous.

@divanshu if the series is arithmetic or geometric, then there are already formulas for the nth term. I use this method when i cant figure out what the formula is.

Answer 32

hey , the question was in my college exam , i can show u paper ??

Answer 33

in fact i see that the question has been asked and answered an hour ago

Answer 34

http://openstudy.com/users/divanshu#/users/divanshu/updates/4e5771260b8b9ebaa8967e3d

Answer 35

@joe that whats i am saying , to apply it on some simple thing about which i can cross check my answers?

Answer 36

ya, i asked it an hour ago and no one answered me completely