Question

Please help I need to heLp my daughter in the morning
. I made a table and on the left side is x values starting with 1 down to 15 and on the right side of the table is y values starting with 1 to 16384. from 1 on the y value side the numbers double example for x value of 1 the y value is 1, for x value 2 y value is 1+2. x value 3, the y value is 1+2+4. I need to write an equation that relates to the x, y. And then it asks to use the equation to find the value of y when x=15. I'm trying to help my daughter and I am stumped.

Answer 1

The sequence generated by the description: 1, 3, 7, 13, 21, 31, ... is somewhat reminiscent of the sequence of Triangular Numbers. [ You can read about them at the following link: http://www.shyamsundergupta.com/triangle.htm ]
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1, 3, 7, 13, 21, 31, ... --> Each term appears to be one more than twice a triangular number.

Moving on, the general term or nth term can be written as the following:
a sub n = n(n-1) + 1 = n^2 - n + 1.

Testing that for n = 1 yields (a sub 1, the first term) = 1^2 - 1 + 1 = 1 which is correct. Other known terms can be checked. The fifth term or (a sub 5) is 5^2 - 5 + 1 = 21.

*The 15th term is (a sub 15) = 15^2 - 15 + 1 = 211. *

As for a formula, I would not use y but a-sub-n because it carries more meaning.

a-sub-n = n^2 - n + 1 where a-sub-n plays the role of what you think of as y and n plays the role of what you think of as x.

Answer 2

@Sculpeyart