The middle bead in a string of 17 beads was the largest and most expensive. Starting from each end, the end beads were $1 each and each bead is $1 more that the one before it, up the middle. If the string beads is worth $100, what is the middle bead worth?
the middle bead is #9. So, you have two AP's of 8 terms each, plus the middle bead. That means that
2(8/2 (2+7))+ x = 100
x = 28
8 1 8
https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
first 8 of arithmetic sequence
a = 1
d = 1
sum from n = 1 to n = 8
Xn = a + d(n-1)
sum =(n/2)[2a+(n-1)d]
= (8/2)[2+7]=72/2
we want twice that to include the other 8
= 72
now that middle one
100 - 72 = 28
To find the value of the middle bead, we need to determine how many beads are on each side of the string.
Let's assume the value of the middle bead is x dollars.
Starting from each end, we know that the value of the end beads is $1 each. Therefore, the sum of the beads on each side of the string (excluding the middle bead) would be ($1 + $2 + $3 + ... + (x-1)).
To find the sum of this arithmetic series, we can use the formula Sn = (n/2)(a + l), where n represents the number of terms, a is the first term, and l is the last term.
The number of terms on each side is (x - 1)/2 since the middle bead is not counted.
So, the sum of the beads on each side of the string is (x - 1)/2 * (1 + (x - 1)) = (x - 1)/2 * x.
Since the total value of the string is $100, the sum of the beads on each side is 100 - x.
Equating the two sums, we have:
(x - 1)/2 * x = 100 - x
Simplifying the equation:
(x - 1)x = 200 - 2x
Expanding:
x^2 - x = 200 - 2x
Rearranging the equation:
x^2 + x - 200 = 0
Now we can solve this quadratic equation for x.
Factoring or using the quadratic formula, we find that x = 14 or x = -15.
Since the bead values are positive, the middle bead is worth $14.
To find the value of the middle bead, we need to determine the total value of all the beads in the string and then subtract the value of the rest of the beads from the total.
The string consists of 17 beads. The value of the first bead is $1 and each subsequent bead is worth $1 more than the previous one. So, we can calculate the total value of the string as the sum of an arithmetic series.
The sum of an arithmetic series can be found using the formula:
Sum = (n/2) * (first term + last term)
Here, n is the number of terms in the series. In this case, n = 17.
The first term is $1 and the last term is the value of the middle bead. Let's say the value of the middle bead is x dollars.
Therefore, the total value of the string can be expressed as:
100 = (17/2) * (1 + x)
Now, we can solve this equation to find the value of x.
Let's simplify the equation:
100 = (8.5) * (1 + x)
Divide both sides by 8.5:
11.76 = 1 + x
Subtract 1 from both sides:
x = 10.76
So, the value of the middle bead is $10.76.