The sum of the 4th and the 14th term of an arithmetic sequence equals 22. The 9th term is 4 larger than the 4th term. Calculate the first term of the sequence.

Question

I've been trying to solve this for like an hour but I just can't figure out how to do it! I'd really appreciate your help! Thanks!!

Answer 1

"The sum of the 4th and the 14th term of an arithmetic sequence equals 22" ----> a+3d + a+13d = 22
2a + 16d = 22
a + 8d = 11 (#1)

"The 9th term is 4 larger than the 4th term" ---> a+8d - (a+3d) = 4
5d = 4
d = 4/5

a = 11-8d
a = 11 - 8(4/5) = 23/5

check:
t4 = 23/5 + 3(4/5) = 35/5 = 7
t9 = 23/5 + 8(4/5) = 55/5 = 11
t14 = 23/5 + 13(4/5) = 75/5 = 15

11+15 = 22
11-7=4

all is good

Answer 2

The third term of an arithmetic sequence is 15, and the fifth term is 37. What is the first term?