# If Sr Denotes the Sum of the First R Terms of an A.P. Then , S3n: (S2n − Sn) is - Mathematics

MCQ

If Sr denotes the sum of the first r terms of an A.P. Then , S3n: (S2n − Sn) is

#### Options

• n

• 3n

• 3

• none of these

#### Solution

Here, we are given an A.P. whose sum of r terms is Sr. We need to find  (S_(3n))/(S_(2n) - S_n).

Here we use the following formula for the sum of n terms of an A.P.

S_n = n/2 [ 2a + (n -1 ) d]

Where; a = first term for the given A.P.

d = common difference of the given A.P.

= number of terms

So, first we find S3n,

S_(3n) = (3n)/2 [ 2a + (3n - 1)d]

=(3n)/2 [2a + 3nd - d ]               .................(1)

Similarly,

S_(2n) = (2n)/2 [ 2a + (2n - 1 ) d ]

= (2n)/2 [2a + 2nd -d]              .................(2)

Also,

S_n = n/2 [ 2a + (n-1) d]

=n/2 [2a + nd - d ]

So, using (1), (2) and (3), we get,

(S_(3n))/(S_(2n) - S_n) = ((3n)/2 [2a + 3nd - d])/((2n)/2 [ 2a + 2nd - d ] - n/2 [ 2a + nd - d ])

Taking n/2 common, we get,

(S_(3n))/(S_(2n) - S_n) =(3[2a + 3nd - d])/(2[2a + 2nd - d ]- [2a  + nd - d])

=(3[2a + 3nd - d])/(4a + 4nd - 2d - 2a  - nd + d)

=(3[2a + 3nd - d])/(2a + 3nd - d)

= 3

Therefore, (S_(3n))/(S_(2n)- S_n )= 3

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Q 17 | Page 58