Strange Sum

Let A₁₇₇₆ be the set { 1, ¹/₂, ¹/₃, ..., ¹/₁₇₇₆ }

Remove any two elements, say a and b, from A₁₇₇₆, and replace them with the single number ab+a+b to form set A₁₇₇₅.
Continue in this manner, until you have performed 1775 such operations, to form set A₁, which contains a single element.

What is this element?

Prove it!

View the solution

Strange Sum Solution

The answer is 1776.

Proof:

Define an operation a**b = ab+a+b

Clearly ** is commutative, because a**b = ab+a+b = ba+b+a = b**a

What's not so obvious is that ** is associative, at least until you realize that

a**b =
(a+1)(b+1)-1

Then (a**b)**c =
((a+1)(b+1))(c+1)-1 = (a+1)((b+1)(c+1))-1 = a**(b**c)

Since ** is associative and commutative, the 1775 operations can be performed in any order, using each of the 1775 operands, and the result will be the same. The number we always arrive at after performing 1775 operations is:

1**(¹/₂)**(¹/₃)** ... **(¹/₁₇₇₆), which is

(²/₁)(³/₂)(⁴/₃)...(¹⁷⁷⁷/₁₇₇₆)-1,
which is equal to 1777-1 = 1776 after everything else cancels.

Oh, You Can't Be Serious!