Question

A club consist of members whose age are in AP. The common difference between the ages is 3 months. If the youngest member is 7 years old and the sum of the ages of all the members is 250, then the number of members in the club is

• A. 15
• B. 25
• C. 20
• D. 30

Answer 1

The Correct Option is B

Let's denote the number of members in the club as 'n'.
We are given that the youngest member is 7 years old, and the common difference between the ages of the members is 3 months, which is equivalent to \(\frac{1}{4}\) years.
So, the ages of the members can be represented as an arithmetic progression (AP):
7, 7\(+\)\(\frac{1}{4}\), 7\(+\)2\((\frac{1}{4})\), 7\(+\)3\((\frac{1}{4})\), …
To find the sum of the ages of all the members, we can use the formula for the sum of an arithmetic series:
Sum =\((\frac{n}{2})\times\)[2a + (n-1)d]
Where:
● n is the number of terms (number of members), 
● a is the first term (age of the youngest member, which is 7 years), 
● d is the common difference \((\frac{1}{4}\) years\()\).
We're given that the sum of the ages of all the members is 250.
Substituting these values into the formula:
250 \(=(\frac{n}{2})\times[2\times7+(n-1)\times(\frac{1}{4})]\)
Now, simplify:
250\(=(\frac{n}{2})\times[14+\frac{(n-1)}{4}]\)
Next, remove the fraction by multiplying both sides by 4:
1000=n\(\times\)[56\(+\)(n-1)]
Now, distribute n on the right side:
1000 = 56n\(+\) n(n-1)
Now, expand and simplify further:
1000 = 56n\(+\)\(n^2-n\)
Combine like terms: 0 =\(n^2+\)55n- 1000 
Now, we need to solve this quadratic equation for n. 
You can factor it or use the quadratic formula: 
\(n=-b\underline+\sqrt\frac{(b2-4ac)}{2a}\)
In this case, a = 1, b = 55, and c =-1000.
\(n=-55\underline+\sqrt\frac{(552-4(1)(-1000))}{2(1)}\)
\(n=-55\underline+\sqrt\frac{(3025+4000)}{2}\)
\(n=-55\underline+\sqrt\frac{7025}{2}\)
\(n=-55\underline+5\sqrt\frac{281}{2}\)
Now, we have two possible solutions: 
1. \(n=-55+5\sqrt\frac{281}{2}\)
2. \(n=-55-5\sqrt\frac{281}{2}\)
Since the number of members in the club should be a positive whole number, wecan ignore the negative solution.
So, the number of members in the club is approximately:
\(N\approx-55+\sqrt\frac{281}{2}\approx25.67\)
Since the number of members must be a whole number, we can round down to the nearest integer.
Therefore, the number of members in the club is 25. 
So, the correct answer is (B): 25.