Question

Find the sum of

(i) all integers between $100$ and $550$, which are divisible by $9$.
(ii) all integers between $100$ and $550$ which are not divisible by $9$.
(iii) all integers between $1$ and $500$ which are multiples of $2$ as well as of $5$.
(iv) all integers from $1$ to $500$ which are multiplies $2$ as well as of $5$.
(v) all integers from $1$ to $500$ which are multiples of $2$ or $5$.

Answer 1

Formula to be used:

(I) $a_n=a+(n-1)d$ or $n=\frac{a_n-a}{d}+1$

(II) $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

(III) $S_n=\frac{n}{2}[a+a_n]$

(i) First number which is divisible by $9$ between $100$ and $550$ is $108$ and last is $549$

Thus, AP is $108,117,...540,549$ with $a=108$ and $d=9$

$\therefore n=\frac{549-108}{9}+1=49+1=50$ [Using I]

Hence, using (III),

Required Sum $=\frac{50}{2}[108+549]=25\times 657=16425$

(ii) Sum of integers which are not divisible by $9$,

Sum of all integers between $100$ and $550(S_{549}-$ Sum of all integers which are divisible by $9$ between $100$ and $550\left(S_{50}\right)$

$\Rightarrow S_{449}=\frac{449}{2}[101+549]=449\times 325=145925$ [Using III]

$\therefore$ Required Sum $=145925-16425$ [From (i)]

$=129500$

(iii) AP formed by those numbers between $1$ and $500$ which are multiples of $2$ as well as $5$ is,

$10,\mathrm{20...480},490$

Here, $a=10,d=10$ and $n=\frac{490-10}{10}+1=49$ [Using I]

Thus, $S_{49}=\frac{49}{2}[10+490]=49\times 250=12250$ [Using III]

(iv) AP formed by those numbers from $1$ to $500$ which are multiples of $2$ as well as of $5$ is,

$10,20,...490,500$.

We can observe that this AP is different from AP of (iii) by only the $500$ term.

Thus, Required Sum $=12250+500=12750$ [From (iii)]

(v) Integers from $1$ to $500$ which are multiples of $2$ are $2,4,\mathrm{6...498},500$

Here, $a=2,d=2$ and $n=\frac{500-2}{2}+1=250$

Thus, $S_{250}=\frac{250}{2}[2+500]=125\times 502=62750$

Integers from $1$ to $500$ which are multiples of $5$ are $5,10,..495,500$

Here, $a=5,d=5$ and $n=\frac{500-5}{5}+1=100$

Thus, $S_{100}=\frac{100}{2}[5+500]=50\times 505=25250$

Required Sum $=S_{250}+S_{100}-12750$ [From (iv)]

$=62750+25250-12750$

$=75250$