Question

Let $x_1,x_2,x_3,...,x_n$ be a sequence of integers such that:
i) -1 $\le$ $x_i$ $\le$ 2 for i = 1, 2 ..., n
ii) $x_1+x_2+x_3+...+x_n$ = 19
iii) $x_1^2+x_2^2+x_3^2+...+x_n^2$ = 99
Determine the minimum and maximum possible values of $x_1^3+x_2^3+x_3^3+...+x_n^3$

• A. 20,99
• B. 19,133
• C. 0,391
• D. 4, 99

Answer 1

The correct option is B 19,133

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, $x_1^3+x_2^3+x_3^3+...+x_n^3$ = -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.