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For each positive integer $n$ consider the set $S_n$ defined as follows: $S_1 = \{1\},\:S_2 = \{2, 3\},\:S_3 = \{4,5,6\}, \: \dots $ and in general, $S_{n+1}$ consists of $n+1$ consecutive integers the smallest of which is one more than the largest integer in $S_n$. Then the sum of all the integers in $S_{21}$ equals to

  1. $1113$
  2. $53361$
  3. $5082$
  4. $4641$
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4 Answers

Best answer
18 votes
18 votes

Every new set $S_{n+1}$ starts after $n$ elements from the starting element of $S_{n}$ . This means that we can find the starting number of $S_{21}$ using Arithmetic progression formula. 

Let $\text{Sum}\left({n}\right)$denote sum of natural numbers until ${n:}$

$S_1$ starts with $= 1$

$S_2$ starts with  $= \text{Sum}\left(1\right) + 1= 2$

$S_3$ starts with $= \text{Sum}\left(2\right) + 1 = \left(1+2\right) + 1 = 4$

$\text{Similarly, }S_{21}$ starts with $S\left(20\right) +1 =  \frac{20(20+1)}{2} + 1 = 211$

Now we need to find sum of ${21}$ consecutive natural numbers starting from ${211}$

Using A.P. sum formula $S_n = \frac{n}{2} \left[2a + \left(n-1\right)*d \right]$ where, $a=$ starting term , $d=$ difference

Sum of elements in $S_{21} =  \frac{21}{2}\left[2\left(211\right) + 20 \right] = \frac{21 * 442}{2} = 4641.$ Option (D) is correct

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4 votes
4 votes

Answer is 4641.

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3 votes
3 votes

S21 = {211,212,213,214,215,...........231}

First element a =211 and Last element l = 231 and total no of elements n = 21 ,so by A.P.

Sum of all the elements of S21 = n/2 [a + l]  = 21/2 * [211 + 231] = 4641

The correct answer is (D) 4641

0 votes
0 votes

The starting no. of $S_{n+1}$  is equal to the sum of the cardinality of all the previous sets plus one, which is equal to the sum of natural nos. upto n plus 1 i.e,

$S_{n+1}=(n(n+1))/2 +1$

$\therefore S_{21}=(20*21)/2 +1$ =211

$S_{21}$ will contain 21 elements from 211 to 231(including).

Now, sum of elements in $S_{21}$= sum of natural nos. upto 231 - sum of natural nos. upto 210

                                                     = (231*232)/2  - (210*211)/2 = 4641

Ans. - D

Answer:

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