Question

Find the sum of all natural numbers which are multiples of $7$ or $3$ or both and lie between$200$ and $500$.

Answer 1

Step 1: Solve for the sum of a multiple of $3$ between $200$ and $500$

The sum is calculated in the following way.

$S=S_3+S_7-S_{21}$where,

$S_3$is the sum of all multiples of $3$ between $200$ and $500$

$S_7$is the sum of all multiples of $7$ between $200$ and $500$

$S_{21}$is the sum of all multiples of $21$ between $200$ and $500$

$S_3$ is an AP with first term $a=201$and common difference $d=3$and last term $a_n=498$

last term of an AP is $a_n=a+(n-1)d$

$$\begin{gathered} \Rightarrow 498=201+(n-1)3 \\ \Rightarrow \frac{297}{3}=n-1 \\ \Rightarrow 99=n-1 \\ \Rightarrow n=100 \end{gathered}$$

Now solving further,

Sum of an AP$=\frac{n}{2}\left(a+a_n\right)$

$$\begin{gathered} \Rightarrow S_3=\frac{100}{2}[201+498] \\ \Rightarrow S_3=50\times 699 \\ \Rightarrow S_3=34950 \end{gathered}$$

Step 2: Solve for the sum of a multiple of $7$ between $200$ and $500$

Similarly, $S_7$ and $S_{21}$ is an AP having the first term $a=203,210$, common difference $d=7,21$and last term $a_n=497,483$ respectively

$a_n$ for $S_7$$=203+(n-1)7$

$$\begin{gathered} \Rightarrow 497=203+(n-1)7 \\ \Rightarrow \frac{294}{7}=n-1 \\ \Rightarrow 43=n \end{gathered}$$

$$\begin{gathered} S_7=\frac{43}{2}(203+497) \\ \Rightarrow S_7=\frac{43}{2}\times 700 \\ \Rightarrow S_7=15050 \end{gathered}$$

Step 3: Solve for the sum of a multiple of $21$ between $200$ and $500$

$a_n$ for $S_{21}=210+(n-1)21$

$$\begin{gathered} 483=210+(n-1)21 \\ \Rightarrow \frac{273}{21}=n-1 \\ \Rightarrow 14=n \end{gathered}$$

$$\begin{gathered} S_{21}=\frac{14}{2}\left(210+483\right) \\ \Rightarrow S_{21}=7\times 693 \\ \Rightarrow S_{21}=4851 \end{gathered}$$

Step 4: Solve for the sum of a multiple of $3$or $7$ between $200$ and $500$

$$\begin{gathered} S=S_3+S_7-S_{21} \\ S=34950+15050-4851 \\ S=45149 \end{gathered}$$

Hence, the sum of all natural numbers which are multiples of $7$ or$3$ or both, and lie between$200$ and $500$ is $45149$.