SOLUTION: Suppose apples are in the shape of a pyramid. On the top layer, there is 1 apple. On the second layer, there are 1+2 apples. On the third layer, there is 1+2+3 apples. On the fourt

Algebra ->  Sequences-and-series -> SOLUTION: Suppose apples are in the shape of a pyramid. On the top layer, there is 1 apple. On the second layer, there are 1+2 apples. On the third layer, there is 1+2+3 apples. On the fourt      Log On


   

Question 1173561: Suppose apples are in the shape of a pyramid. On the top layer, there is 1 apple. On the second layer, there are 1+2 apples. On the third layer, there is 1+2+3 apples. On the fourth layer, there 1+2+3+4 apples. If there are N number of layers, how many apples are there in total?
Found 4 solutions by ikleyn, Boreal, ewatrrr, greenestamps:
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

The number of balls in the k-th layer from the top is  %28k%2A%28k%2B1%29%29%2F2, k = 1, 2, 3, . . . , n.


The number of balls in the entire triangle pyramid of n layers is  %28n%2A%28n%2B1%29%2A%28n%2B2%29%29%2F6.    ANSWER


This formula can be easily proved by the method of Mathematical induction.


For more details, see this link

https://math.stackexchange.com/questions/1148720/finding-the-formula-for-the-number-of-golf-balls-in-a-triangular-pyramid-with-n


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Be aware (!)


1)   The formula in the post by @Boreal is INCORRECT (!)


2)   The formula in the post by  @ewatrrr     sum%28%28N%5Bi%5D%28N%5Bi%5D-1%29%2F2%29%2C+i=1%2C+N%29%29       is incorrect,  too  (!)


      The correct writing is  THIS     sum%28+%28i%28i%2B1%29%2F2%29%2C+i=1%2C+N%29%29,

      and it gives that final compact formula as presented in my post.


. . . . . . . .

@ewatrrr fixed her post after seeing my solution.



Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
The formula for this is N(N+1)/2.
there are a variety of ways to show this. I look at the average of all the numbers and multiply by n.
The average is (1/2)(n+1), and that is multiplied by n.
The proof came to me looking at lane markers on a track course.

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 
Hi,
Pyramid of Apples
On the top layer, there is 1 apple. On the second layer, there are 1+2 apples. On the third layer, there is 1+2+3 apples. On the fourth layer, there 1+2+3+4 apples.
top  1 
then 3  
then 6
then 10 (4th layer)  N%28N%2B1%29%2F2+=+10
Each Layer has %28N%28N%2B1%29%29%2F2+ Apples
If there are N number of layers, how many apples are there in total?
That would be the Sum of the Layers
  
N   N%28N%2B1%29%2F2%29     Multiplier             Tot
1	1		     1                    1
2	3   	             4/3                  4          
3	6		     5/3                 10
4	10		     6/3                 20

Multiplier is (N+2)/3  

SUM =  = %28N%28N%2B1%29%2F2%29%28%28N%2B2%29%2F3%29
Wish You the Best in your Studies.


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The number of apples in layer k is

1+2+3+...+(k-1)+k

This sum is equal to

(k(k+1))/2

That expression is equal to

C(k+1,2)

So the sum we are looking for is

C(2,2)+C(3,2)+C(4,2)+...+C(N,2)+C(N+1,2)

Using the hockey stick identity for Pascal's Triangle, this sum is equal to

C%28N%2B2%2C3%29+=+%28%28N%2B2%29%28N%2B1%29%28N%29%29%2F3%21

Do an internet search if you aren't familiar with the hockey stick identity....