Question

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?

Answer 1

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

Answer 2

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