# A square pyramid of empty cans containing 5525 cans is made.The formula for a square pyramid with n layers is 1^2+2^2+3^2....+n= n(n+1)(2n+1) < all divided by 6. Use this formula to determine...

A square pyramid of empty cans containing 5525 cans is made.

The formula for a square pyramid with n layers is 1^2+2^2+3^2....+n= n(n+1)(2n+1) < all divided by 6. Use this formula to determine the number of layers in the pyramid of cans.

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The total number of cans = 5525

The top layer = 1 can

the second layer = 2^2 = 5

the third layer = 3^2 = 9

.....

then nth layer = n^2

==: 1^2 + 2^2 + 3^2 + ....+n^2 = 5525

==: 1 + 4 + 9 + 16 + ...+ n^2 = 5525

The sum of the aeries is:

S = n(n+1)(2n+1)/6 = 5525

Multiply by 6:

==> n(n+1)(2n+1) = 33,150

==> 2n^3 + 3n^2 + n= 33,150

==> 2n^3 + 3n^2 + n - 33,150 = 0

==> (n-25)(2n^2 + 53n + 1326) = 0

==> n= 25

**Then there are 25 layers of cans **

The number of layers n in the pyramid is n.

Then the number n is decided by the inequation 1+2^2+3^2.....n^2 <= 5525.

We know that the sum 1+2^2+3^2+...+n^2 = n(n+1)(2n+1)/6.

Therefore we have to solve for n in n(n+1)(2n+1)/6 = 5225.

n(n+1)(2n+1) = 5225*6 =31350

For n = 25, n(n+1)(2n+1) 25*26*51 = 33150

For = 26, n(n+1)(2n+) = 37206.

Therefore the number of layers in the pyramid = n = 25 .