Question

Let a sequence have 200 zeroes to start with. In step 1, to every position in the sequence we add 1. In step 2 to every position which is a multiple of 2 in the sequence we add 1. In step 3, to every position which is a multiple of 3, we add 1. This is continued up to ${200}^{th}step.$. After the 200th step, what will be the value in the 128th position?

• A. $8$
• B. $16$
• C. $128$
• D. $200$

Answer 1

The correct option is B $16$

$S_n=0+0+0+0+0+0+0+0+0+0+0+.....200times\left(i\right)S_n=1+1+1+1+1+1+1+1+1+1+......200times\left(ii\right)S_n=1+2+1+2+1+2+1+2+1+2+......\left(iii\right)S_n=1+2+2+2+1+3+1+2+2+2+....\left(i\right)S_n=2+3+3+3+2+4+2+3+3+....\left(ii\right)S_n=2+4+3+4+2+5+2+4+3+.....\left(iii\right)S_n=2+4+4+4+2+6+2+4+4+...$

The ${200}^{th}$ step is the $2^{nd}$ step

The ${128}^{th}$ position is a multiple of $2$ hence it varies according to the other multiple of $2$ values

$1,2,2,3,4,4,5,6,6,7,8,8,9,..$

After every $3n$ steps the gap between the step and the value is $3n-n=2n$

After $198$ steps i.e. $n=66$ steps

the value is $2n=132$

${199}^{th}$ step: $133$

${200}^{th}$ step: $134$

Number at ${128}^{th}$ position is $134$