what is the first term of an arithmetic sequence if the 7th term is 21 and the 10th term is 126

Answer 1

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) = 126-21
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)=21-245=-224

T(n) = -224 + 35n

Answer 2

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:
t₇ = -189 + 6(35) = 21
t₁₀ = -189 + 9(35) = 126