Given: The sum of the first k terms of a series S is given as Sk=3k2+5k. Concept: To find the nth term of a series, we use the formula: an=Sn−Sn−1 If the nth term forms an arithmetic progression (AP), the common difference (d) is given by: d=an+1−an Calculation: We are given Sk=3k2+5k. ⇒an=Sn−Sn−1 Also Sn=3n2+5n and Sn−1=3(n−1)2+5(n−1) ⇒an=[3n2+5n]−[3(n−1)2+5(n−1)] ⇒an=[3n2+5n]−[3(n2−2n+1)+5n−5] ⇒an=[3n2+5n]−[3n2−6n+3+5n−5] ⇒an=3n2+5n−3n2+6n−3−5n+5 ⇒an=6n+2 The series is an arithmetic progression (AP) if the difference between consecutive terms is constant. ⇒ Common difference d=an+1−an ⇒an+1=6(n+1)+2=6n+6+2=6n+8 ⇒d=(6n+8)−(6n+2)=6 ∴ The terms of the series form an arithmetic progression with a common difference of 6 .