Question

Find the sum of:
1. The first $50$ multiples of $11$
2. The multiples of $7$ between $0$ and $1000$
3. The integers between $1$ and $100$ which are not divisible by $3$ and $7$.

Answer 1

$1)$ First $50$ multiples of $11$

$S_{50}=\frac{n}{2}[2a+(n-1\left)d\right]=\frac{50}{2}[2\times 11+(50-1)\times 11]=25[22+49\times 11]=25[22+539]=25+561=14025$

$2)$ The sum multiples of $7$ between $0$ and $1000$

Adding all multiples of $7$ from $7$ to 974 (last multiple of $7$in the given range)

So,

$7+14+....+994=7(1+2+...+\frac{994}{7})=7(1+2+....+142)=7\frac{\left(142\right)\left(143\right)}{2}=71071$

$3)$ Sum of number divisible by $3=1683$

Sum of number divisible by $7=735$

Sum of number divisibleby $21=210$

$\therefore$ Sum of numbe not divisible by $3\&7=$ Sum of first $100$ naturalnumber $-$ Sum of division of $3-$ Sum of division of $7+-$ Sum of divisor of $21$

=5050-1685-735+210

=2842