Question

If the natural numbers are divided into group of $\left\{1\right\},\{2,3\},\{4,5,6\},\{7,8,9,10\}...$ then the sum of $50th$ group is

• A. $62525$
• B. $65225$
• C. $56255$
• D. $55625$

Answer 1

Answer: (A)

$62525$

$1^{st}$ group contains $1$ number

$2^{nd}$ group contains $2$ numbers

$3^{rd}$ group contains $3$ numbers

.

.

.

.

${50}^{th}$ group contains $50$ numbers

Total no. of numbers$=1+2+\mathrm{3.....50}$

Sum of n natural no.s$=\frac{n(n+1)}{2}$

For $n=50,s=\frac{50\times 51}{2}=1275$

For $n=49,s=\frac{49\times 50}{2}=1225$

Sum of ${50}^{th}$ group$=$sum upto ${50}^{th}$group $-$sum upto ${49}^{th}$ group

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

Sum upto ${50}^{th}$ group $\to$

$n=1275$

$a=1$

$d=1$

Sum $=\frac{1275}{2}[2+12+4]=813450$

Sum upto ${49}^{th}$ group$\to$

$n=1225$

$a=1$

$d=1$

Sum$=\frac{1225}{2}[2+1224]=750925$

Sum of ${50}^{th}$ group$=813450-750925$

$=62525$.