Can you sum the parts within ten seconds?

[Ian]

I asked this of my maths lecturer in uni. He took about ten seconds to mentally perform the calculation/integration. Can you beat his time?

Q.
A cyclist rides 100 miles from point A to point B at a constant velocity of 20mph. As he leaves point A, a bee on his handlebars flies ahead of him toward point B at a steady velocity of 25mph. When the bee arrives at point B it immediately returns to meet the cyclist somewhere in-between. The bee then flies at the same velocity in-between point B and the cyclist until the cyclist reaches point B.
What distance does the bee fly in total? (you have ten seconds)

 

[Jimmy Snyder]

I asked this of my maths lecturer in uni. He took about ten seconds to mentally perform the calculation/integration. Can you beat his time?

Q.
A cyclist rides 100 miles from point A to point B at a constant velocity of 20mph. As he leaves point A, a bee on his handlebars flies ahead of him toward point B at a steady velocity of 25mph. When the bee arrives at point B it immediately returns to meet the cyclist somewhere in-between. The bee then flies at the same velocity in-between point B and the cyclist until the cyclist reaches point B.
What distance does the bee fly in total? (you have ten seconds)

Spoiler

When the cyclist reaches point B, 5 hours will have passed. In those 5 hours the bee will have traveled 125 miles.

There's an interesting anecdote that goes with this puzzle. It was asked of the Mathematician John Von Neumann who pondered for a few moments and answered it. The puzzler said "Oh, yo u know the trick." and Von Neumann replied "No, I summed the series.". I had tried to sum the series when I was in High School and didn't know how to do it. However, when I heard the anecdote, (and after I learned how to do it), I tried again to sum the series and found it rather easy. Of course, the trick answer is even easier.

 

[CylonMath]

Spoiler

120?

 

[DyslexicHobo]

The bee then flies at the same velocity in-between point B and the cyclist until the cyclist reaches point B.

I'm not sure I understand what this means. The bee travels the same velocity as the cyclist until the cyclist reaches point B?

I agree with CylonMath's answer, but Jimmysnyder is usually very accurate in his problem-solving abilities, so I'm doubting my answer.

 

[Borek]

The bee flies all the time at 25 mph.

 

[Jimmy Snyder]

I'm not sure I understand what this means. The bee travels the same velocity as the cyclist until the cyclist reaches point B?

You (and the OP) mean speed, not velocity. I took the OP to mean that the bee traveled at 25 mph at all times. The infinite acceleration at the turning points, the infinite number of turns at the end of the flight, and the 5 hours of flight at 25mph indicate to me that this bee is a better mathematician than physicist.

 

[regor60]

Just imagine the kind of shape this bee is in after all those windsprints