Prove that there is a positive integer that equals the sum of the positive integers not exceeding it. Is your proof constructive or nonconstructive?

Constructive Proof

Step-by-step explanation:

Let x be a positive integer

x must be equal to sum of all positive integers exceeding it

i.e.

x = x + (x - 1) + ( x - 2) + + 2 + 1

Equivalently,

x = ∑i (where i = 1 to x)

The property finite sum;

∑i (i = 1 to x) = x(x + 1)/2

So,

x = x(x + 1)/2 Multiply both sides by 2

2 * x = 2 * x(x + 1)/2

2x = x(x + 1)

2x = x² + x subtract 2x from both sides

2x - 2x = x² + x - 2x

0 = x² + x - 2x Rearrange

x² + x - 2x = 0

x² - x = 0 Factorise

x(x - 1) = 0

So,

x = 0 or x - 1 = 0

x = 0 or x = 1 + 0

x = 0 or x = 1

But x ≠ 0

So, x = 1

The statement is only true for x = 1

This makes sense because 1 is the only positive integer not exceeding 1

1 = 1

It is a Constructive Proof

A proof is constructive when we find an element for which the statement is true.

g(x) is the negative form of f(x)? because whenever its the negative form it reflects its self over the x axis. idk if this is right though.

step-by-step explanation:

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step-by-step explanation: