Let x1,x2,x3,...,xn be a sequence of integers such that:
i) -1 ≤ xi ≤2 for i = 1, 2 ..., n
ii) x1,x2,x3,...,xn = 19
iii) x21,x22,x23,...,x2n = 99
Determine the minimum and maximum possible values of x31,x32,x33,...,x3n
Option (b)
Solution:
Let a, b and c denote the number of -1s, 1s and 2s in the sequence respectively. So,
-a + b + 2c = 19 and a + b + 4c = 99 (neglecting the zeroes)
So, a = 40 – c and b = 59 – 3c where 0 ≤ c ≤ 19 (as b ≥ 0)
So, x31,x32,x33,...,x3n = -a + b + 8c = 19 + 6c
When c = 0 (a = 40, b = 59), the minimum value is 19; when c = 19 (a = 21, b =2), maximum value is 133.