[superprismatic]

The numbers 1, Y, and Y^2, Y a positive

integer, are used as the basis for a

sequence in which every value after the

first three is the average of the

previous three numbers. This is

continued indefinitely and the sequence

is found to converge to the value 321.

Determine Y.

SUPERPRISMATIC NOTE: Not all numbers

in this sequence are integers! So,

programmers beware!

[HoustonHokie]

Y = 25

[Guest]

The first few terms of the series are:

1

x

x²

(x² + x + 1) / 3

(4x² + 4x + 1) / 9

(16x² + 7x + 4) / 27

It is trivial to observe that the third term (x²) must be > 321 and the second term (x) must be less than 321

Also, the 4th and 5th terms must be smaller than 321 and the 6th term must be greater than 321 in order to have a rolling average around 321

Otherwise the average will be either < 321 or > 321

Then,

x² > 321

x > 17

(x² + x + 1) / 3 < 321

x < 32

(4x² + 4x + 1) / 9 < 321

(2x+1)2 < 2889

x < 26

(16x2 + 7x + 4) / 27 > 321

(4x+2)² - 9x > 27*321

(4x+2)² - 9x > 8667

(4x+2)² - 9x > (93.something)²

So,

4x + 2 > 94

x > 23

But, x > 23 may not be exact as the square on the other side was > 93 and there is a term of -9x

So, if x = 24, we have

94² - 9*24 = 8620 which is less than 8667

So x > 24

Now we have 24 < x < 26

Since x is an integer, x must be 25

Edited August 6, 2009 by DeeGee

[superprismatic]

The first few terms of the series are:

1

x

x²

(x² + x + 1) / 3

(4x² + 4x + 1) / 9

(16x² + 7x + 4) / 27

It is trivial to observe that the third term (x²) must be > 321 and the second term (x) must be less than 321

Also, the 4th and 5th terms must be smaller than 321 and the 6th term must be greater than 321 in order to have a rolling average around 321

Otherwise the average will be either < 321 or > 321

Then,

x² > 321

x > 17

(x² + x + 1) / 3 < 321

x < 32

(4x² + 4x + 1) / 9 < 321

(2x+1)2 < 2889

x < 26

(16x2 + 7x + 4) / 27 > 321

(4x+2)² - 9x > 27*321

(4x+2)² - 9x > 8667

(4x+2)² - 9x > (93.something)²

So,

4x + 2 > 94

x > 23

But, x > 23 may not be exact as the square on the other side was > 93 and there is a term of -9x

So, if x = 24, we have

94² - 9*24 = 8620 which is less than 8667

So x > 24

Now we have 24 < x < 26

Since x is an integer, x must be 25

Nice job! I like your analysis. It's easy to follow. I wish I had thought of this way to do it.