The cross-section of a honeycomb can be thought of as a central regular hexagon surrounded by a ring of 6 or more...

Question

The cross-section of a honeycomb can be thought of as a central regular hexagon surrounded by a ring of 6 or more hexagons, surrounded by a ring of 12 or more hexagons, and so on. Assuming the central hexagon is ring 1, the ring of 6 hexagons as ring 2 (and so on), determine the total number of hexagons in the first 50 rings. 

Can you please include the process of how to solve the problem?

 

https://i.imgsafe.org/cce3b7e28b.png (The web link is a diagram to help solve for the problem)

Answer 1

First, having a picture/diagram is pretty important for visual learners; studying it helps your comprehension.

Second, listing a few terms in the sequence will help to determine what kind of sequence it is and to determine what formulas to use for finding a specific term or the sum of the terms so far. Note these definitions:
       A Sequence is a list of things (usually numbers) that are in order.
       A Partial Sum is a sum of part of a sequence.

Now, you may appreciate “coloring” or otherwise counting the number of hexagons in each ring using honeycomb (hexagon) graph paper:
  http://www.thealmightyguru.com/Humor/Inventions/Images/Grid-Hex.gif

For this honeycomb, the terms (the number of regular hexagons in each ring) are:
         1, 6, 12, 18, 24, 30, 36, … [PLZ determine this for yourself !]

Except for the first term, this is an arithmetic sequence with this definition:
       a₁ = 6
       a_n = a₁ + 6(n-1) for n>2

The Partial Sum of an Arithmetic Sequence with initial term a₁ and common difference d is given by:
       S_n = n(a₁+a_n)/2

For our honeycomb, that is (considering the 1 again):
      1 + S₄₉ = 1 + 49(6+a₄₉)/2
                 = 1 + 49(6+(6+6(49-1))/2                     [insert value for a_n]
                 = 7351

Note: You should try calculating a₅=24 and S₅=61 (for example) to gain confidence that these calculations are correct. In one minute, I created an Excel spreadsheet with the sequence and a SUM formula. 7351 it is !