Section 1.1: The Method of Difference of Squares

To tackle this problem, we will utilize the concept of the difference of squares.

For demonstration, picture a square with an area of ( a^2 ) and another with an area of ( b^2 ).

When we subtract the area ( b^2 ) from ( a^2 ), we create a specific shape.

Next, we can break this shape into two distinct rectangles.

By connecting sides of equal length ( a - b ), we arrive at a rectangle featuring side lengths of ( a + b ) and ( a - b ).

This leads us to the formula for the difference of two squares. We can express the general term of the series in terms of ( k ).

Next, we rationalize the term as shown below.

It’s worth noting that we can apply the difference of squares to the denominator of our term.

Upon further simplification, we arrive at the following expression.

Now, applying the expression for the general term to our series yields:

Notice that the middle terms cancel out, leaving us with just the very first and last terms.

And that concludes our answer.

Section 1.2: Challenge Accepted!

How impressive is that? What was your reasoning during this exercise? I’d love to hear your thoughts in the comments below!

The first video, Can You Solve This Puzzle in Under 5 Minutes?, presents similar challenges that will test your problem-solving abilities.

The second video, Learn to Solve a Rubik's Cube in Under 5 Minutes, offers a great tutorial for mastering the Rubik's Cube quickly.