SOLUTION: 1,2,2,4,3,8,7,10 what is next number?

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Question 983310: 1,2,2,4,3,8,7,10 what is next number?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
A very common recursion formula is of this form:

t%5Bn%2B1%5D=+a%2An%2At%5Bn%5D%2Bb%2An%2Bc, where a,b, and c are constants:

Substitute n=1

t%5B1%2B1%5D=+a%2A1%2At%5B1%5D%2Bb%2A1%2Bc
t%5B2%5D=+a%2A1%2A1%2Bb%2Bc
2=+a%2Bb%2Bc

Substitute n=2

t%5B2%2B1%5D=+a%2A2%2At%5B2%5D%2Bb%2A2%2Bc
t%5B3%5D=+2a%2A14%2B2b%2Bc
24=+28a%2B2b%2Bc

Substitute n=3

t%5B3%2B1%5D=+a%2A3%2At%5B3%5D%2Bb%2A3%2Bc
t%5B4%5D=+3a%2A24%2B3b%2Bc
66=+72a%2B3b%2Bc

We will find a,b, and c from this system:

system%2816a%2Bb%2Bc=14%2C28a%2B2b%2Bc=24%2C72a%2B3b%2Bc-66%29

Subtracting the 1st equation from the 2nd equation

12a%2Bb=10

Subtracting the 2nd equation from the 3rd equation

44a%2Bb=42

So now we have this system:

system%2812a%2Bb=10%2C44a%2Bb=42%29

Subtracting the first equation from the second:

32a=32

a=1

Substitute in 12a+b=10

12(1)+b=10
   12+b=10
      b=-2

Substitute in 16a+b+c=14

16(1)+(-2)+c=14
      16-2+c=14
        14+c=14
           c=0

So recursion formula:

t%5Bn%2B1%5D=+1%2An%2At%5Bn%5D%2B%28-2%29%2An%2B0

t%5Bn%2B1%5D=+n%2At%5Bn%5D-2n

Now we test the recursion formula to see if t%5B5%5D=256

t%5B4%2B1%5D=+4%2At%5B4%5D-2%2A4

t%5B5%5D=+4%2A66-8

t%5B5%5D=+256

So this is the correct recursion formula.

So we use it to find t%5B6%5D

t%5B5%2B1%5D=+5%2At%5B5%5D-2%2A5

t%5B6%5D=+5%2A256-10

t%5B6%5D=+1270   <--answer

Edwin