Number of ways in which three numbers in AP can be selected from 1,2,3, . .,n is

Question

Number of ways in which three numbers in AP can be selected from 1,2,3, . .,n is

Number of ways in which three numbers in AP can be selected from 1,2,3, . .,n is
A. `((n-1)/(2))^(2)`, if n is even
B. `(n(n-2))/(4)`, if n is even
C. `((n-1)^(2))/(4)`, if n is odd
D. `(n(n+1))/(2)`, if n is odd.

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

Related Questions

The sum of three numbers is 6 . When second number is subtracted from thrice the sum of first and third number , we get number 10. Four times the thir

2 Answers 4 Views

If the number of ways of selecting n cards out of unlimited number of cards bearing the number 0,9,3, so that they cannot be used to write the number

2 Answers 5 Views

A number of four different digits is formed with the help of the digits 1,2,3,4,5,6,7 in all possible ways. (i) How many such numbers can be formed? (

2 Answers 4 Views

A lady gives a dinner party to 5 guests to be selected from nine friends. The number of ways of forming the party of 5, given that two of the friends

2 Answers 4 Views

A man has three friends. The number of ways he can invite one friend everyday for dinner on six successive nights so that no friend is invited more th

2 Answers 4 Views

In how many ways can a team of 6 horses be selected out of a stud of 16, so that there shall always be three out of A B C A B C, but never A A, B B or

2 Answers 4 Views

Statement-1: A number of four different digits is formed with the help of the digits 1,2,3,4,5,6,7 in all possible ways. The number of ways which are

2 Answers 4 Views

Statement-1: The total number of ways in which three distinct numbers in AP, can be selected from the set {1,2,3, . .,21}, is equal to 100. Statement-

2 Answers 4 Views

Three numbers are in continued proportion, whose mean proportional is 12 and the sum of the remaining two numbers is 26, then find these numbers.

2 Answers 4 Views

If x,y,z be three positive prime numbers. The progression in which `sqrt(x),sqrt(y),sqrt(z)` can be three terms (not necessarily consecutive) is

2 Answers 4 Views

Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - B::C
If a,b,c and in AP, then a+c=2b
a and b both are odd or both are even.
Case I if n is even
Let n=2m in which m are evenn annd m are odd numbers.
`therefore`Number of ways`=.^(m)C_(2)+.^(m)C_(2)=2*.^(m)C_(2)=2*(m(m-1))/(2)`
`=(n)/(2)((n)/(2)-1)=(n(n-2))/(4)" "[becausen=2m]`
Case II If n is odd
Let n=2m+1 in which m are even and m+1 are odd numbers.
`therefore`Number of ways`=.^(m)C_(2)+.^(m+1)C_(2)`
`(m(m-1))/(2)+((m+1)m)/(2)=m^(2)=((n-1)^(2))/(4)" "[becausen=2m+1]`