What is the sum of all the common terms between the given series S1 and S2 ?

S1 = 2, 9, 16, .........., 632

S2 = 7, 11, 15, .........., 743

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  1. 6974
  2. 6750
  3. 7140
  4. 6860

Answer (Detailed Solution Below)

Option 1 : 6974
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Detailed Solution

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GIVEN:

Two series given i.e. S1 and S2

FORMULA USED:

an = a + ( n - 1 ) d 

Sn = n/2 [2a + (n - 1) d ] 

Where,

a= nth term in the sequence , n= number of terms , a = first term in sequence, d = common difference , Sn = Sum 

CALCULATION:

Here, given series S1  and S2 are in A.P.

So, series will move by adding a fixed common difference ( second term - first term) in consecutive terms 

S1 = 2 , 9 , 16 , 23, 30 , 37 , 44 , 51 ,........ 632      [As  here d = 7 .So, add 7 in previous term to get next term]

S2  = 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51 , ......... 743  [ Here d = 4 ] 

Now, let us take a third series S3 which is common series                    [It will contain common numbers of both series only]

So, from S1 and S2 series, we have 1st common term = 23 , 2nd common term = 51 , d = ( 51 - 23) = 28 

So, add 28 in second term to get third term and so on

S3 = 23 , 51 , .................. \(\le\)  632              [ As, 632 is less than 743 so common between them should be less than 632]

Now, we have a = 23 , d = 28 

⇒ an = a + ( n - 1) d    \(\le\) 632 

⇒ 23 + ( n - 1) × 28 \(\le\) 632 

⇒ ( n -  1) × 28  \(\le\)  ( 632 - 23 ) 

⇒ ( n - 1) × 28  \(\le\) 609 

⇒ n -1 \(\le\) 609 /28 

⇒ n - 1 \(\le\) 21.75 

⇒ n \(\le\) 22 .75

 As, n should be equal to or less than 22.75 . So, take n = 22 

Now, as we know that 

Sn = n/2 [ 2a + ( n - 1) d ] 

⇒ Sn = 22/2 [ 2 × 23 + ( 22 - 1) 28]  = 11 [46 + 21 × 28 ] 

⇒ 11 [ 46 + 588 ] = 11 × 634  = 6974

Hence, Sum of all the common terms of the series are 6974 .

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