Recover terms from partial sums and compare a geometric tail

(a) A sequence has partial sums $S_n=n^2+6n$.

(i) Find and simplify $u_n$.3 marks

(ii) Show it is an AP.2 marks

(iii) Find $u_1+u_3+\cdots+u_{49}$.2 marks

(b) A GP has nonzero first term a and nonzero ratio r with $-1<r<1$. Its (n−1)-th term equals the sum after, excluding, its n-th term.

(i) Show $r^2+r-1=0$.3 marks

(ii) Deduce r.1 mark