If the second term of an arithmetic sequence is -2 and the fourth term is 6, find the seventh term
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
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 ...
To find the seventh term of an arithmetic sequence, we need to determine the common difference (d) between the terms.
The common difference (d) can be found by subtracting the second term from the first term, or by subtracting the fourth term from the third term since the difference between consecutive terms remains constant in an arithmetic sequence.
Let's calculate the common difference:
d = second term - first term
= -2 - first term
d = fourth term - third term
= 6 - (-2)
= 6 + 2
= 8
We now know that the common difference (d) is 8.
To find the seventh term, we can use the formula for the nth term of an arithmetic sequence:
tn = a + (n - 1)d
Where:
tn = nth term
a = first term
n = position of the term
d = common difference
Since we are looking for the seventh term, n = 7.
We are given the second term as -2, so a = -2.
Substituting these values into the formula:
t7 = -2 + (7 - 1) * 8
t7 = -2 + 6 * 8
t7 = -2 + 48
t7 = 46
Therefore, the seventh term of the arithmetic sequence is 46.