the sum of the 4th and 8th terms is 10, and the sum of the 6th and 12th terms is 22. Find the first three terms.
a + 3d + a + 7d = 10
2a + 10d = 10
a + 5d = 5
a+5d + a+11d = 22
2a + 16d = 22
a + 8d = 11
subtract them
3d = 6
d = 2
back into the first
a + 5d = 5
a + 10 = 5
a = -5
first 3 terms are -5, -3, -1
To find the first three terms, we need to set up a system of equations using the given information.
Let's assume that the common difference between the terms is represented by 'd', and the first term is represented by 'a'.
The 4th term can be written as a + (4-1)d = a + 3d. The 8th term can be written as a + (8-1)d = a + 7d.
According to the given information, the sum of the 4th and 8th terms is 10:
(a + 3d) + (a + 7d) = 10
Simplifying this equation, we get:
2a + 10d = 10
Similarly, the 6th term can be written as a + (6-1)d = a + 5d. The 12th term can be written as a + (12-1)d = a + 11d.
According to the given information, the sum of the 6th and 12th terms is 22:
(a + 5d) + (a + 11d) = 22
Simplifying this equation, we get:
2a + 16d = 22
Now we have a system of equations:
2a + 10d = 10
2a + 16d = 22
We can solve this system of equations using any method, such as substitution or elimination. Let's use the elimination method to solve it.
Multiply the first equation by 8 and the second equation by 5 to eliminate the 'a' term:
16a + 80d = 80
10a + 80d = 110
Subtract the first equation from the second equation:
(10a + 80d) - (16a + 80d) = 110 - 80
-6a = 30
a = -5
Substitute the value of 'a' into one of the equations:
2(-5) + 10d = 10
-10 + 10d = 10
10d = 10 + 10
10d = 20
d = 20/10
d = 2
Therefore, the first term (a) is -5, and the common difference (d) is 2.
To find the first three terms, we can substitute 'a' and 'd' into the arithmetic sequence formula:
1st term = a = -5
2nd term = a + d = -5 + 2 = -3
3rd term = a + 2d = -5 + 2(2) = -5 + 4 = -1
So, the first three terms of the sequence are: -5, -3, -1.