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

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)   (Show Source): You can put this solution on YOUR website!
.

The number of balls in the k-th layer from the top is , k = 1, 2, 3, . . . , n. The number of balls in the entire triangle pyramid of n layers is . 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

/\/\/\/\/\/\/

Be aware (!)

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

2)   The formula in the post by @ewatrrr            is incorrect, too (!)

      The correct writing is THIS     ,

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

. . . . . . . .

@ewatrrr fixed her post after seeing my solution.

Answer by Boreal(15235)   (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)   (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) Each Layer has Apples If there are N number of layers, how many apples are there in total? That would be the Sum of the Layers N 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 = = Wish You the Best in your Studies.

Answer by greenestamps(13200)   (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

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

RELATED QUESTIONS
A pyramid of blocks is built in layers. The bottom layer has 6 to the 2 or 36 blocks.... (answered by ikleyn)
There are 8436 steel balls, each with a radius of 1 centimeter, stacked in a pile, with 1 (answered by josgarithmetic)
A very bored grocer was stacking oranges one day. She decided to stack them in a... (answered by venugopalramana)
Build a sugar cube pyramid as follows: First make a 5 x 5 x 1 bottom layer. Then center a (answered by ikleyn)
GREAT PYRAMID OF ORANGES A very bored grocer was stacking oranges one day. She decided (answered by ikleyn,greenestamps)
A pile of logs has 24 in the fist layer,23 in the second layer,22 in the third layer,and... (answered by checkley71)
a grocer makes a display of canned tomatoes that are on sale there are 24 cans in the... (answered by josmiceli)
A pile of logs contains 20 logs on the topmost layer and each lower layer contains one... (answered by greenestamps)
a stack of logs has 52 logs on the bottom layer. each subsequent layer has 10 fewer logs... (answered by ameanes,solver91311)