Question

Calculate the sum of odd numbers between 300 and 450 which are divisible by 15.
A. 1,875
B. 3,710
C. 3,302
D. 3,375

Answer 1

Hint: We will start by finding the numbers which are divisible by 15 and are in between 300 and 450 and are odd. Then we will use the formula for finding the sum of A.P. with first term a and last term l to find the sum.

Complete step-by-step answer:
Now, we have to calculate the sum of odd numbers between 300 and 450 which are divisible by 15.
Now, we will start by finding the numbers which are divisible by 15 and are in between 300 and 450. So, we have them as,
315, 330, 345,………435.
Now, since we have to take only odd numbers. Therefore, we will remove all the numbers which are divisible by 2. So, we have,
315, 345, 375,……435.
Now, we have an A.P. with first term a = 315
The common difference of the A.P = d = 30
Now. We have the number of terms in A.P using the formula for finding the last term on an A.P i.e.
$\begin{align}
  & l=a+\left( n-1 \right)d \\
 & 435=315+\left( n-1 \right)30 \\
 & 435-315=\left( n-1 \right)30 \\
 & 120=\left( n-1 \right)30 \\
 & \dfrac{120}{30}=n-1 \\
 & 4+1=n \\
 & 5=n \\
\end{align}$
Now, we know that the sum of the terms of an A.P. with first term a and last term l is,
$S=\dfrac{n}{2}\left( a+l \right)$
Now, we have,
$\begin{align}
  & n=5 \\
 & a=315 \\
 & l=435 \\
 & \Rightarrow Sum=S=\dfrac{5}{2}\left( 315+435 \right) \\
 & =\dfrac{5}{2}\left( 750 \right) \\
 & =5\times 375 \\
 & =1875 \\
\end{align}$
Hence, the correct option is (A).

Note:To solve this question we have used the fact that the sum of the terms of an A.P. with first term as a and last term l is $\dfrac{n}{2}\left( a+l \right)$. Also, it is important to note that odd numbers are those which are not divisible by 2.