Question

Prove that Sn -S (n-1)= tn

Answer 1

Please see the attachment

Answer 2

This relation can be proved by using an AP

Let an AP with first term as a and common difference as d.

Sₙ = n/2 (2a + (n-1)d) = an + nd(n-1)/2                        ---------> (1)

Sₙ₋₁ = (n-1)/2 *(2a + (n-1-1)d ) = a(n-1) + (n-1)(n-2)d/2   ----------->(2)

Subtracting 2 from 1 gives

Sₙ - Sₙ₋₁ =  an + nd(n-1)/2 -  [ a(n-1) + (n-1)(n-2)d/2]

Sₙ - Sₙ₋₁ = a(n - (n-1)) + nd(n-1)/2 - (n-1)(n-2)d/2

Sₙ - Sₙ₋₁ = a + d/2*(n-1)*( n - (n-2) )

Sₙ - Sₙ₋₁ = a + d/2*(n-1)*2

Sₙ - Sₙ₋₁ = a + d*(n-1)

Sₙ - Sₙ₋₁ = a + (n-1)d

Sₙ - Sₙ₋₁ = Tₙ