Question

How many multiples of $9$ lie between $10$ and $300$?

Answer 1

Step 1. List of numbers:

The numbers $18,27,36,45,54,\dots ,297$ are multiples of $9$ lie between$10$ and $300$.
Here, $27-18=36-27=9$
So, the list of numbers $18,27,36,45,54,\dots ,297$ forms an AP, whose first term (a) = $18$, and common difference (d) = $9$.

Step 2. Finding the value of $\mathit{n}$:

Let there are $n$ terms in the above sequence, then
$a_n=297$
The formula for Arithmetic Progression AP is
$\begin{array}{cc}\Rightarrow & a+(n-1)d=a_n\\ \Rightarrow & a+(n-1)d=297\\ \Rightarrow & 18+(n-1)9=297\\ \Rightarrow & 18+9n-9=297\\ \Rightarrow & 9+9n=297\\ \Rightarrow & 9n=297-9\\ \Rightarrow & 9n=288\\ \Rightarrow & n=\frac{288}{9}\\ \Rightarrow & n=32\end{array}$

Therefore, the value of $nis32.$