What number is a number that is even and not even at the same time? And what are they divided in half?

Definitions:

A number, x, is even if it satisfies: x = 2n for some integer n

A number, x, is odd if it satisfies: x = 2k+1 for some integer k

Rewriting the Question:

In this case the question is asking if there is a number, x, that is has the following property:

x = 2n for some integer n AND x = 2k+1 for some integer k

Proof:

We can set these two equations equal:

2n = x = 2k + 1

2n = 2k + 1

2n - 2k = 1

2(n - k) = 1

Let number y = n - k. Note that since n and k are integers, n-k (and therefore, y) must also be an integer.

2y = 1

You can see on the left side (2y) that this becomes the definition of an even number! So the left side is an even number and the right side is 1. Since we know by definition that 1 is not an even number and k must be an integer (not 0.5), this equation becomes false.

As a result, there are NO numbers that can be both even AND odd.

Part A.

Option 1: linear function

Option 2: exponential function

This is because for Option 1 we can readily see that the money is constantly increasing by 100 every year hence linear. While for Option 2 the increase changes every year hence exponential.

Part B.

Option 1: Linear function has general form of:

y = mx + b

where y is the amount of money while x is the number of years

Calculating for slope m by using the data points in year 1 and 2 ( you can use any 2 data pairs)

m = (y2 – y1) / (x2 – x1)

m = (1200 – 1100) / (2 – 1)

m = 100

While b is the value of y when x = 0, hence b = 1000

Therefore:

y = 100 x + 1000

Option 2: Exponential function has general form of:

y = y0 (1+r)^x

where y0 is the initial amount and r is the growth rate

Solving for r using data at time x = 1

1100 = 1000 (1 + r)^1

r = 0.1

Therefore:

y = 1000 (1.1)^x

Part C. Solving for y for each options when x = 20

Option 1:

y = 100 (20) + 1000 = 3000

Option 2:

y = 1000 (1.1)^20 = 6727.50

Therefore Belinda should invest her money in Option 2 since it has greater return after 20 years.