Question
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
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
Answer: Option B
:
B
Option (b)
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.
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:
B
Option (b)
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.
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