First slide

Arithmetic progression

Question

A club consists of members whose ages are in A.P., the common difference being 3 months. If the youngest member of the club is just 7 years old and the sum of the ages of all the members is 250 years, then the number of members in the club are

Moderate

A

15
B

25
C

20
D

30

Solution

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
Here a = 1st term = 7 years, d = 3 months = $\frac{1}{4}$ year,
$S_{n} = 250$
$\begin{array}{l} ∴ 250 = \frac{n}{2} [2 \times 7 + (n - 1) \times \frac{1}{4}] \\ \Rightarrow 250 = \frac{n}{2} (\frac{n + 55}{4}) \\ \Rightarrow 2000 = n^{2} + 55 n \\ \Rightarrow n^{2} + 55 n - 2000 = 0 \\ \Rightarrow (n - 25) (n + 80) = 0 \Rightarrow n = 25 . \end{array}$
$∴$ Number of members in the club = 25.

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