If the second term of an arithmetic sequence is -2 and the fourth term is 6, find the seventh term

Answer 1

You should know the basic formulas for these.
"the second term of an arithmetic sequence is -2" ---> a+d = -2
"the fourth term is 6" ---> a+5d = 6

subtract the two equations, that will give you d
sub back into the first equation to get a
then find a+6d

Answer 2

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence an is given by:

an = a1 + ( n - 1 ) d

a2 = a1 + ( 2 - 1 ) d

a2 = a1 + 1 ∙ d

a2 = a1 + d

- 2 = a1 + d Subtract d to both sides

- 2 - d = a1 + d - d

- 2 - d = a1

a1 = - 2 - d

a4 = a1 + ( 4 - 1 ) d

a4 = a1 + 3 ∙ d

a4 = a1 + 3 d

6 = a1 + 3 d Subtract 3 d to both sides

6 - 3 d = a1 + 3 d - 3 d

6 - 3 d = a1

a1 = 6 - 3 d

a1 = a1

- 2 - d = 6 - 3 d Add 3 d to both sides

- 2 - d + 3 d = 6 - 3 d + 3 d

- 2 + 2 d = 6 Add 2 to both sides

- 2 + 2 d + 2 = 6 + 2

2 d = 8 Divide both sides by 2

d = 8 / 2 = 4

a1 = - 2 - d

a1 = - 2 - 4

a1 = - 6

Now:

an = a1 + ( n - 1 ) d

a7 = a1 + ( 7 - 1 ) d

a7 = a1 + 6 d

a7 = - 6 + 6 ∙ 4

a7 = - 6 + 24

a7 = 18

By the way, your arithmetic sequence:

an = - 6 + ( n - 1 ) ∙ 4

- 6 , - 2 , 2 , 6 , 10 , 14 , 18 , 22 ...