if the 4th term of an arithmetic sequence is 9 and the 8th term is 19
To find the common difference of the arithmetic sequence, we can use the formula for finding the nth term of an arithmetic sequence:
nth term = a + (n - 1)d
Where:
- nth term is the term number we want to find
- a is the first term of the sequence
- d is the common difference between consecutive terms
- n is the number of terms
Let's use this formula to find the common difference (d) of the arithmetic sequence.
Given:
4th term = 9
8th term = 19
We can plug these values into the formula to form two equations:
9 = a + (4 - 1)d -- Equation 1
19 = a + (8 - 1)d -- Equation 2
Simplifying the equations, we get:
9 = a + 3d
19 = a + 7d
Now, we can solve this system of equations using any method (substitution, elimination, etc.). For simplicity, let's use the substitution method.
From Equation 1, let's solve for a in terms of d:
a = 9 - 3d
Substitute this expression for a in Equation 2:
19 = (9 - 3d) + 7d
19 = 9 + 4d
10 = 4d
Divide both sides by 4:
d = 10/4
d = 2.5
So, the common difference (d) for this arithmetic sequence is 2.5.
Now, if you want to find the first term (a), you can substitute the value of d into Equation 1 or Equation 2.
Using Equation 1:
9 = a + 3(2.5)
9 = a + 7.5
a = 9 - 7.5
a = 1.5
Therefore, the first term (a) of the arithmetic sequence is 1.5.
so, the 6th term is midway between the 4th and the 8th: 14
I guess that's not what you wanted to know. Note that
T8 = T4 + 4d
Now you can find d, then a.