Question

Question 2 (iii)
Find the sum of those integers from 1 to 500 which are multiples of 2 or 5.

Answer 1

Since, multiples of 2 or 5 = Multiple of 2 + Multiple of 5 – Multiple of LCM (2,5) i.e. 10.
$\therefore$ Multiples of 2 or 5 from 1 to 500
= List of multiple of 2 from 1 to 500 + List of multiple of 5 from 1 to 500 - List of multiple of 10 from 1 to 500
= (2, 4, 6, ... , 500) + (5, 10, 15, ... ,500) - (10, 20, ... ,500)

Each of the above series form an AP.

Let the number of terms in first series be $n_1$.
$So,500=2+(n_1-1)2[n^{th}termofanAP=a+(n-1\left)d\right]]$
$\Rightarrow 498=(n_1-1)2$
$\Rightarrow n_1-1=249$
$\Rightarrow n_1=250$

Let the number of terms in the second list be $n_2$.
$So,500=5+(n_2-1)5\Rightarrow 495=(n_2-1)5$
$\Rightarrow 99=(n_2-1)\Rightarrow n_2=100$

Let the number of terms in the third list be $n_3$.
$So,500=10+(n_3-1)10\Rightarrow 490=(n_3-1)10\Rightarrow \frac{490}{10}=n_3-1\Rightarrow n_3=50$

Sum of multiples of 2 or 5 from 1 to 500
= Sum of (2,4,6 … 500) + Sum of (5, 10, ….500) – Sum of (10,20,….500)
$=\frac{n_1}{2}[2+500]+\frac{n_2}{2}[5+500]-\frac{n_3}{2}[10+500][\because S_n=\frac{n}{2}(a+l\left)\right]$
$=\frac{250}{2}\times 502+\frac{100}{2}\times 505-\frac{50}{2}\times 510$
$=250\times 251+505\times 50-25\times 510=62750+25250-12750$
$=88000-12750=75250$