1. If the combined mean of \(n\) quantities is \(\bar{x}\), and the sum of the first \((n-1)\) quantities is \(k\), then the \(n\)th quantity will be —

(a) \(k+n\) (b) \(\bar{x}+k\) (c) \(\bar{x}-k\) (d) \(n\bar{x}-k\)

2. If the mean of the numbers \(x_1, x_2, x_3, x_4, ..., x_n\) is \(\bar{x}\), then the value of \((x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + ... + (x_n - \bar{x})\) will be —

(a) 0 (b) 1 (c) 3 (d) 5

3. If the middle number of the first \((2n + 1)\) consecutive natural numbers is \(\frac{n + 103}{3}\), then find the value of \(n\).

4. If the average of the numbers \(7, x - 3, x + 3, 10, x - 5\) is \(15\), then their median will be \(10\).

5. If the average of \( (p + q) \) numbers is \( x \), and the average of \( p \) of those numbers is \( y \), then the average of the remaining \( q \) numbers will be _____ .

6. If the average of the first four numbers out of five is 26, and the average of the last four numbers is 25, then find the difference between the first and the last number.

7. If the average of the \(n\) numbers \(x_1, x_2, x_3, ..., x_n\) is \(\bar{x}\), then the average of \(Kx_1, Kx_2, Kx_3, ..., Kx_n\) is —— (where \(K \ne 0\)).

8. If the combined average of the numbers 7, \(x - 3\), 10, \(x + 3\), and \(x - 5\) is 15, then the median is _____.

(a) 16 (b) 10 (c) 18 (d) 24

9. If the average of \(n\) natural numbers is \(\cfrac{n+8}{4}\), then find the value of \(n\).

(a) 4 (b) 6 (c) 8 (d) 10

10. If the average of \(n\) natural numbers is \(\cfrac{n+8}{4}\), then what is the value of \(n\)?

(a) 4 (b) 6 (c) 8 (d) 0

11. If the average of the first ten natural numbers is A and the median is M, then the relationship between A and M is —

(a) \(A\gt M\) (b) \(A\lt M\) (c) \(A=\cfrac{1}{M}\) (d) \(A=M\)

12. Translate: If the average of \(p\) numbers is \(b\), and the average of \((p + q)\) numbers is \(a\), then the average of the \(q\) numbers is _____

13. If the average of the numbers 6, 7, x, 8, y, and 14 is 9, then

(a) x+y=21 (b) x+y=19 (c) x-y=21 (d) x-y=19

14. If the average of the numbers 6, 7, x, 8, y, and 16 is 9, then—?

(a) x + y = 21 (b) x + y = 17 (c) x - y = 21 (d) x - y = 19

15. If \(n\) is an even number, then the median will be the average of the \(\left(\cfrac{n}{2}\right)\)-th and \(\left(\cfrac{n}{2}−1\right)\)-th observations.

16. If the product of three consecutive positive geometric numbers is 64, then the mean proportional between the first and the third is ______.

17. If the numbers \((2 + x)\), \((8 + x)\), and \((26 + x)\) are in continued proportion, then what is the value of \(x\)?

(a) 4 (b) 3 (c) 2 (d) 1

18. If the fourth of 5 consecutive geometric numbers is 54 and the fifth is 162, then find the first number.