A collection of dimes is arranged in a triangular array with 14 coins in the base row, 13 in the next, 12 in the next, and so forth. Find the value of the collection.

number of dimes

= 1+2+3+...+13+14
= 7(1+14) = 105

value = 105(.10) = $10.50

To find the value of the collection, we first need to determine the pattern of the triangular arrangement.

Looking at the arrangement, we can observe that there are a total of 14 + 13 + 12 + ... + 1 rows. This is an arithmetic series where the first term is 14, the last term is 1, and the common difference is -1 (each row has one fewer coin).

We can calculate the sum of an arithmetic series using the formula:

S = (n/2)(a + l),

where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.

In this case, n = 14 (since there are 14 terms), a = 14, and l = 1.

Using the formula, we can find the sum:

S = (14/2)(14 + 1) = 7 * 15 = 105.

Therefore, there are a total of 105 dimes in the collection.

To find the value of the collection, we need to multiply the number of dimes by the value of each dime.

Since each dime is worth 10 cents, the value of the collection is:

105 dimes * 10 cents/dime = 1050 cents = $10.50.

Therefore, the value of the collection is $10.50.

To find the value of the collection, we need to determine the number of dimes in each row and then sum up the total value.

The base row has 14 dimes.
The next row has 13 dimes.
The row after that has 12 dimes.
And so on.

To find the total number of dimes, we need to calculate the sum of the numbers from 14 to 1. This can be done using the formula for the sum of an arithmetic series.

The sum of an arithmetic series is given by the formula: S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.

In this case, the first term is 14, the last term is 1, and the number of terms is 14. Plugging these values into the formula, we get:

S = (14/2)(14 + 1)
S = 7 * 15
S = 105

So, the total number of dimes in the collection is 105.

The value of a dime is $0.10. Therefore, to find the total value of the collection, we need to multiply the total number of dimes by $0.10:

Total value = 105 * $0.10
Total value = $10.50

Therefore, the value of the collection is $10.50.