how do i solve the sum of the first two terms of an arithmetic sequence is 16. the sum of the second and third term is 28. determine the first three terms, algebraically
To determine the first three terms of an arithmetic sequence algebraically, we need to set up a system of equations using the given information.
Let's assume that the first term of the sequence is "a" and the common difference is "d".
Step 1: Given that the sum of the first two terms is 16, we can write the equation:
a + (a + d) = 16
Step 2: Given that the sum of the second and third terms is 28, we can write the equation:
(a + d) + (a + 2d) = 28
Now, let's solve the system of equations step-by-step.
Step 3: Simplify the equations:
2a + d = 16
2a + 3d = 28
Step 4: Eliminate variable "a" by multiplying the first equation by 2 and subtracting it from the second equation:
(2a + 3d) - (2a + d) = 28 - 32
2a + 3d - 2a - d = 12
2d = 12
Step 5: Solve for d:
2d = 12
d = 12 / 2
d = 6
Step 6: Substitute the value of d back into the first equation and solve for a:
2a + 6 = 16
2a = 16 - 6
2a = 10
a = 10 / 2
a = 5
Step 7: Calculate the first three terms:
The first term (a) is 5.
The second term (a + d) is 5 + 6 = 11.
The third term (a + 2d) is 5 + 2(6) = 5 + 12 = 17.
Therefore, the first three terms of the arithmetic sequence are 5, 11, and 17.
To determine the first three terms of an arithmetic sequence algebraically, we can set up a system of equations using the given information.
Let's denote the first term as "a" and the common difference as "d".
1. The sum of the first two terms is 16:
The sum of the first two terms in an arithmetic sequence can be expressed as:
a + (a + d) = 16
2. The sum of the second and third terms is 28:
The sum of the second and third terms in an arithmetic sequence can be expressed as:
(a + d) + (a + 2d) = 28
We now have a system of two equations with two variables. We can solve this system to find the values of "a" and "d", which will allow us to determine the first three terms of the arithmetic sequence.
Let's solve it algebraically:
From equation 1, we have:
2a + d = 16
From equation 2, we have:
2a + 3d = 28
To eliminate the "a" term, we can subtract equation 1 from equation 2:
(2a + 3d) - (2a + d) = 28 - 16
This simplifies to:
2d = 12
Dividing both sides of the equation by 2, we get:
d = 6
Substituting the value of "d" back into equation 1, we have:
2a + 6 = 16
Simplifying this equation, we get:
2a = 10
Dividing both sides of the equation by 2, we get:
a = 5
Now we know that the first term "a" is 5 and the common difference "d" is 6.
To determine the first three terms, we can substitute these values into the formula for the terms of an arithmetic sequence:
First term: a = 5
Second term: a + d = 5 + 6 = 11
Third term: a + 2d = 5 + 2(6) = 5 + 12 = 17
Therefore, the first three terms of the arithmetic sequence are 5, 11, and 17.
a1 + a2 = 16
a2 + a3 = 28
and since it is an arithmetic sequence
a2 - a1 = a3 - a2
or
a2 = (a1+a3)/2 (the average of course)
substitute back
a1 + (a1+a3)/2 = 16
(a1+a3)/2 + a3 = 28
solve those two equations for a1 and a3, then go back and get a2