Question

In an arithmetic sequence of terms , Sn represents sum of n terms , then what is Sn - $S_{n- 1}$

Answer 1

Sn represents the sum of n terms .
we know,
Sum of n terms = n/2{2a + (n-1)d}
where a is first term and d is the common difference of AP.
Sn = n/2{2a + (n-1)d}

again,
Sn-1 is the sum of (n+1)terms
so, Sn-1 = (n-1)/2{2a+(n-1-1)d}
=(n-1)/2{2a+(n-2)d}

now,
Sn - Sn+1 = n/2{2a+(n-1)d} - (n-1)/2{2a+(n-2)d}
= 1/2[ 2a(n-n+1) + d{n²-n-n²+3n-2}]
= 1/2[2a + d(2n-2)]
= a + d(n-1)
= a + (n-1)d

but we know,
Tn (nth term) = a + (n-1)d
so, we observed,
Sn - Sn-1 = Tn