Question

The value of the sum in the ${50}^{th}$ bracket of $\left(1\right)+(2+3)+(4+5+6)+(7+8+9+10)+\dots$ is

• A. $62525$
• B. $65225$
• C. $56255$
• D. $555625$

Answer 1

The correct option is A $62525$

In the given series, the $rth$ term in the series is the sum of $r$ terms.
1st term is $1$.
2nd term is $2+3$
3rd term is $4+5+6$ and so on.
If we take the first terms of every term, they form a sequence $1,2,4,7,\mathrm{11...}$
Suppose the first term of the $50th$ term is $F_{50}$
So we get $F_2-F_1=1$
$F_3-F_2=2$
$F_4-F_3=3$
.
.
$F_{50}-F_{49}=49$
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Adding all we get $F_{50}-F_1=1+2+3+.....49$

$\Rightarrow F_{50}=1+\frac{49\times 50}{2}$

$\Rightarrow F_{50}=1226$

Hence, the value in the $50th$ bracket is $(1226+1227+...+1275)$ (50 terms)

Sum $=\frac{50}{2}[2\times 1226+(50-1\left)1\right]$

$=25\times 2501$

$=62525$