Math/Precal
posted by Nancy on .
what is the first term of an arithmetic sequence if the 7th term is 21 and the 10th term is 126

For an arithmetic sequence, the nth term is given by:
T(n)=T(0)+kn,
where k is a constant.
Knowing T(7)=21, and T(10)=126
we find k using
T(10)T(7) = 12621
T(0)+10k  (T(0)+7k) = 105
3k = 105
k=35
From the value of k, we find T(0):
T(n)=T(0)+35n
T(7)=T(0)+35(7) = 21
T(0)=21245=224
T(n) = 224 + 35n 
7th term is 21 > a+6d = 21
10th term is 126 > a + 9d = 126
subtract them:
3d = 105
d = 35
back in a+6d = 21 > a = 189
first term is 189
check:
t_{7} = 189 + 6(35) = 21
t_{10} = 189 + 9(35) = 126