Question

The number of shot in a triangular pile is greater by 150 than half the number of shot in a square pile, the number of layers in each being the same ; find the number of shot in the lowest layer of the triangular pile.


✔️✔️ quality answer needed✔️✔️ ​

Answer 1

This is an arithmetic sequence: 1,2,3,4... [counting down from the top of the pile]

The nth term of an arithmetic sequence is

a(n) = a(1) + (n-1)d where a(1) = the 1st term, d = the common difference

In this case a(n) = 1 + (n-1) = n

The sum of the 1st n terms of the sequence is S(n) = (n/2)(a(1) + a(n)) = 105

So S(n) = (n/2)(1+n) = 105

Solve for n:

n^2 + n - 210 = 0

Factor:

(n-14)(n+15) = 0

Take the positive solution, n=14

So there are 14 layers

Answer 2

Sn = n/2 ( 2a (n-1) d ) = 150

n^2 + n - 210 = 0

(n -14) (n +15)

Therefore the correct answer is 14