Question

A club consists of members whose ages are in AP the common difference being 6 months. The youngest member of the club is just 14 years old and the sum of the ages of all the members is 500 years. Find the number of members in the club

Answer 1

common difference=6/12year=1/2year
first term=14years
now use AP sum formula
Sn=n/2 {2a+(n-1) d}
500=n/2 {14 x 2+(n-1) 1/2}
500=14n+n^2/4-n/4
500=55n/4+n^2/4
n^2+55n-2000=0
now use quadratic formula
n=25,-80
but n=-80 isn't possible
so,
n=25

Answer 2

Answer:

25

Step-by-step explanation:

We are given that A club consists of members whose ages are in AP

Common difference = d = 6 months = $\frac{6}{12} =\frac{1}{2}$  years

The youngest member of the club is just 14 years old

So, first term of A.P. = a= 14

Now we are given that  the sum of the ages of all the members is 500 years.

Formula of sum of first n terms = $S_n=\frac{n}{2} (2a+(n-1)d)$

where n is the number of terms in A.P.

So, $500=\frac{n}{2} (2(14)+(n-1) \times \frac{1}[2])$

$1000=n(28+(n-1) \times 0.5)$

$1000=n(28+(0.5n-0.5))$

$1000=n(27.5+0.5n)$

$1000=27.5n+0.5n^2$

$27.5n+0.5n^2-1000=0$

$0.5(n-25)(n+80)=0$

$n=25,-80$

Since n cannot be negative

So, total number of members in the club is 25