Question

The fourth term of an arithmetic sequence is 5 and the eighth term is 21, find the first term

Answer 1

To find the first term of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1)d

where:
a_n is the nth term
a_1 is the first term
d is the common difference between consecutive terms

Given that the fourth term is 5 and the eighth term is 21, we can set up two equations:

5 = a_1 + 3d (1)
21 = a_1 + 7d (2)

Subtracting (1) from (2), we get:

21 - 5 = a_1 + 7d - (a_1 + 3d)
16 = 4d
d = 4

Substitute d = 4 into equation (1):

5 = a_1 + 3(4)
5 = a_1 + 12
a_1 = -7

Therefore, the first term of the arithmetic sequence is -7.