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The sum of n, 2n, 3n terms of an A.P. are S1 , S2 , S3 respectively. Prove that S3 = 3(S2 – S1 ) - CBSE Class 10 - Mathematics

ConceptSum of First n Terms of an AP

Question

The sum of n, 2n, 3n terms of an A.P. are S1 , S2 , S3 respectively. Prove that S3 = 3(S2 – S1 )

Solution

Let a be the first term and d be the common difference of the given A.P. Then,

S1 = Sum of n terms

⇒ S_1 = \frac { n }{ 2 } {2a + (n – 1)d} ….(i)

S_2 = Sum of 2n terms

⇒ S_2 = \frac { 2n }{ 2 } [2a + (2n – 1) d] ….(ii)

and, S3 = Sum of 3n terms

⇒ S_3 = \frac { 3n }{ 2 } [2a + (3n – 1) d] ….(iii)

Now, S2 – S1

= \frac { 2n }{ 2 } [2a + (2n – 1) d] – \frac { n }{ 2 } [2a + (n –1) d]

S_2 – S_1 = \frac { n }{ 2 } [2 {2a + (2n – 1)d} – {2a + (n – 1)d}]

= \frac { n }{ 2 } [2a + (3n – 1) d]

\∴ 3(S_2 – S_1 ) = \frac { 3n }{ 2 } [2a + (3n – 1) d] = S_3  [Using (iii)]

Hence, S3 = 3 (S2 – S1 )

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Solution The sum of n, 2n, 3n terms of an A.P. are S1 , S2 , S3 respectively. Prove that S3 = 3(S2 – S1 ) Concept: Sum of First n Terms of an AP.
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