The sum to t of an Arithmatic Sequence is 1/2 t(5t-1).Determine the k term
S_n=\frac{n}{2}( a_1 + a_n)=\frac{n}{2}[ 2a_1 + (n-1)d].
Sum= n/2 (2a1 + (n-1)d)
you are looking for n.
1/2 (t(5t-1))=t/2 (2a1+(t-1)d)
t(5t-1)=t(2a1 + td-d)
5t-1=2a1-d + dt
d=5 a1=2 works
Sn= k/2(a1+an)
1/2 (k(5k-1))=k/2(2+an)
5k-1=2+an
an= 5k+1
check: let k=3
terms are 2, 7, 12, sum is 21
Sn=1/2*3*14=21
checks.