Question

If the set of natural numbers is partitioned into subset $S_1$ = {1},$S_2$= {2,3},$S_3$= {4,5,6} and so on.Then the sum of the terms is $S_{50}$ is_______.

• A. 62525
• B. 25625
• C. 62500
• D. 25600

Answer 1

The correct option is A

62525

From symmetry, we observe that $S_{5}0$ has 50 terms.First terms of $S_1$,$S_2$,$S_3$,$S_4$,........... are 1,2,4,7........ Let $T_n$ be the first term of $n^{th}$ set.Then

S=$T_1$ + $T_2$ + $T_3$ +...................+ $T_n$

$\Rightarrow$ S=1+2+4+7 +11+ ............+ $T_{n-1}$ + $T_n$

or S = 1+2+4+7+ ............+ $T_{n-1}$ + $T_n$

Therefore on subtracting

0 = 1+ [1+2+4+ ............+ $T_n$ + $T_{n-1}$] - $T_n$

or 0 = 1 + $\frac{n(n-1)}{2}$ - $T_n$ $\Rightarrow$ $T_n$ = 1 + $\frac{n(n-1)}{2}$

$\Rightarrow$ $T_{50}$ = First Term in $S_{50}$ = 1226 and common difference = 1

Therefore sum of the terms in

$S_{50}$ = $\frac{50}{2}$ $2\times 1226\times (50-1)\times 1$

=25(2452 + 49) = 25(2501) = 62525