find n and tn

Question

t1=20, d=-3/2 and Sn=108 1/2

Answer 1

t_n=t₁+(n-1)d=20-3(n-1)/2=(43-3n)/2.

S_n=108½=217/2.

S_n=t₁+t₂+t₃+...+t_n=(43-3)/2+(43-6)/2+...=43n/2-(3/2)(1+2+...+n)=43n/2-(3/2)n(n+1)/2.

43n/2-(3/2)n(n+1)/2=217/2,

86n-3n(n+1)=434,

83n-3n²=434,

3n²-83n+434=0=(n-7)(3n-62), so n=7 (because n must be an integer).