GeekDad Puzzle of the Week Solution: Pumpkin Picking Problems

Welcome back, puzzle solvers. We're still waiting for John to get back with the last few pumpkins. As one of the puzzle entries commented, "Mike is a jerk." Of course he is, but before we prove that let's recap last weeks puzzle ...
Puzzle Recap:

Every year at GeekDad Puzzle Central we grow our own pumpkins. This year we decided 50 pumpkins would be plenty for carving, baking and chunkin'. I was assigned 2 interns, Mike and John, to help with the planting, cultivating, and harvesting. This week's puzzle is based on the lopsided deal Mike made with John. Solve this one and you could find yourself with a $50.00 ThinkGeek gift certificate.

On planting day Mike struck a deal with John. Mike would do the planting and John would pick the pumpkins in the fall. Being a bit short-sighted, John agreed to the deal.

Mike planted the first seed, then planted the second seed 1 yard away from the first one. Mike increased the distance by two yards between each subsequent seed. So it is 3 yards from the 2nd seed to the 3rd seed, 5 yards from the 3rd seed to the 4th seed, and so on until all 50 seeds are planted in a long row.

The interns care for the pumpkins all summer long.

Harvesting day comes and John finds himself with a big task at hand. The pumpkins grew so big that John could only carry one at a time. The storage bin is next to the first pumpkin and John has to walk out to each pumpkin, pick it, and then walk it back to the storage bin to place it. You can safely assume that the pumpkins grew exactly where the seeds were planted and, yes, we have a huge backyard.

Can you figure out the total distance John walked to get all 50 pumpkins into the storage bin?

Why Mike is a jerk:

Let's look at John's journey as he gathers the pumpkins by listing the first few distances:

*Distance from pumpkin 1 to the bin = 0
Distance from pumpkin 2 to the bin = 1
Distance from pumpkin 3 to the bin = 1 + 3 = 4
Distance from pumpkin 4 to the bin = 1 + 3 + 5 = 9
Distance from pumpkin 5 to the bin = 1 + 3 + 5 + 7 = 16
*
There's a pattern here: the distance from each pumpkin to the bin is a perfect square. So John's journey is the sum of squares times 2. Well there's a formula for that:

Setting n to 49 in the above formula's gives us 40,425 yards. Multiply that by 2 and you get a final answer of 80,850 yards which is just shy of 46 miles (45.9375 miles for the accuracy geeks). That's why Mike is a jerk.

All that's left now is to reveal the winner of this week's $50.00 ThinkGeek gift certificate. That lucky person is John Martin. For everyone else, here's a code for $10.00 off a $50.00 or more order at ThinkGeek starting November 1st.