Question

A club consists of few members who's ages are in AP, the common difference being 6 months . the youngest member of the club is just 10 years old and the sum of the ages of all the members is 400 years .find the number of members in the club

Answer 1

The sum of an AP is S = (n//2)(2a + (n-1)d)

Let the number of members in the club be N.

As the ages of the members are in AP then,

• the first term of an AP(a) is the age of the youngest member of the club = 10
• common difference(d) = 6 months = 1/2 years

Therefore,
the sum of their ages S= (N/2)(2*10 + (N-1)(1/2))
Also S= 400 (given)

On equating, we have

400 = (N / 2)( 2*10 + ( N - 1 )( 1 / 2 ))
400 = (N / 4)( 40 + N - 1)
1600 = N^2 + 39N
N^2 + 39N - 1600=0
(N - 25)(N + 64) = 0
N = 25 or N = -65

As N is the number of members in the club, it must be a non-negative number. So, N=25

The number of members in the club = 25


To check,
put N=25 in the following equation,
S = (N / 2)( 2*10 + ( N - 1 )( 1 / 2 ))
we got
S= (25 / 2) ( 20 + 24*(1/2))
S= (25 / 2) ( 20 + 12)
S= (25 / 2) (32)
​​​​​​​S= 25*16
S= 400 (as given)
So checked.