Question

if SN denotes the sum of n terms of an AP whose common difference is D and first term is a. find SN - 2 Sn - 1 + Sn - 2

Answer 1

Given a is first term and d be the common difference.

Sn = (n/2)[ 2a + ( n -1) d]

Now Sn - 2Sn-1 + Sn +  2

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

⇒ (1/2)[ 2an + n( n -1) d] + [ 4a(n - 1) + 2(n - 1)( n - 2) d] +[ 2a(n + 2)+ ( n + 1) (n + 2) d]

⇒ (1/2)[ 2a[ n - 2n + 2 + n + 2] + d [ n2 - n - 2n2 + 6n - 4 + n2 + 3n + 2] ]

⇒ (1/2)[ 2a(4) + d(8n - 2) ]

= [ 4a + (4n - 1)d]

Answer 2

Given a is first term and d be the common difference.

Sn = (n/2)[ 2a + ( n -1) d]

Now Sn - 2Sn-1 + Sn +  2

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

⇒ (1/2)[ 2an + n( n -1) d] + [ 4a(n - 1) + 2(n - 1)( n - 2) d] +[ 2a(n + 2)+ ( n + 1) (n + 2) d]

⇒ (1/2)[ 2a[ n - 2n + 2 + n + 2] + d [ n2 - n - 2n2 + 6n - 4 + n2 + 3n + 2] ]

⇒ (1/2)[ 2a(4) + d(8n - 2) ]

= [ 4a + (4n - 1)d]