SOLUTION: Five hundred and twenty two circular cylinders are stacked in rows in such a way that the number in a row is one less than in the row below. The bottom row contains 32 cylinders.

Question 1006085: Five hundred and twenty two circular cylinders are stacked in rows in
such a way that the number in a row is one less than in the row below.
The bottom row contains 32 cylinders. How many rows are there?
How many cylinders are there in the top row?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!

you have two formulas to work with.

Sn = n * (A1 + An) / 2

An = A1 + (n-1) * d

Sn is the sum of the terms in the arithmetic series.
An is the nth term of the arithmetic series.
n is the number of terms in the arithmetic series.
d is the commpon difference.
A1 is the first term in the arithmetic series.

the first term in the series is the bottom row.

A1 = 32

the common difference is -1 because each succeeding row agove the first row has one less barrel in it.

d = -1

The sum of the terms in the arithmetic series is 522 barrels.

Sn = 522

Sn = n * (A1 + An) / 2 becomes:

522 = n * (A1 + An) / 2

multiply both sides of this formula by 2 to get:

1044 = n * (A1 + An)

replace A1 with 32 to get:

1044 = n * (32 + An) *****

hold on to this for now.

An = A1 + (n-1) * d becomes:

An = 32 + (n-1) * -1 which becomes:

An = 32 -n + 1 which becomes:

An = 33 - n *****

the two formulas you have to work with now are:

1044 = n * (32 + An)
An = 33 - n

replace An in the first equation with 33 - n to get:

1044 = n * (32 + 33 - n)

distribute the multiplication to get:

1044 = 32n + 33n - n^2

combine like terms to get:

1044 = 65n - n^2

add n^2 to both sides of the equation and subtract 65n from both sides of the eqaution to get:

n^2 - 65n + 1044 = 0

factor this quadratic to get:

(n-36) * (n-29) = 0

solve for n to get:

n = 36
n = 29

you have two possible solutions for n.

substitute in the An equation to get:

An = 33 - n

when n = 36, An = -3

since you can't have -3 barrels, this solution is no good.

when n = 29, An = 4

this solution looks good so we'll test it out in the Sn equation.

Sn = n * (A1 + An) / 2 becomes:

522 = 29 * (32 + 4) / 2 which becomes:

522 = (29 * (36) / 2 which becomes:

522 = 522 which confirms the solution of n = 29 and An = 4 is good.

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