Find the arithmetic sequence if the fourth term is -6 and the eleventh term is -34
Why did the arithmetic sequence go to therapy? Because it had some serious problems with addition and subtraction! Now, let's solve this one together.
In an arithmetic sequence, each term is obtained by adding a constant difference (d) to the previous term.
Let's say the first term of the sequence is "a" and the common difference is "d".
The fourth term, which we'll call "a₄", can be expressed as: a₄ = a + (4-1)d
Similarly, the eleventh term, which we'll call "a₁₁", can be expressed as: a₁₁ = a + (11-1)d
We are given that a₄ = -6 and a₁₁ = -34. Plugging these values in:
-6 = a + 3d (Equation 1)
-34 = a + 10d (Equation 2)
Now, let's solve these two equations simultaneously to find the values of "a" and "d".
Subtracting Equation 1 from Equation 2, we get:
-34 + 6 = a + 10d - (a + 3d)
-28 = 7d
Dividing both sides by 7:
d = -4
Now that we have the value of "d", we can substitute it back into Equation 1 to find the value of "a":
-6 = a + 3(-4)
-6 = a - 12
Adding 12 to both sides:
6 = a
So, the first term of the arithmetic sequence is 6 and the common difference is -4.
Therefore, the arithmetic sequence is: 6, 2, -2, -6, -10, -14, -18, -22, -26, -30, -34.
To find the arithmetic sequence, we will use the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n-1)d
where aₙ represents the nth term, a₁ represents the first term, n represents the term number, and d represents the common difference between consecutive terms.
We are given that the fourth term (n=4) is -6 and the eleventh term (n=11) is -34.
Step 1: Substitute the given values into the formula to create two equations:
For the fourth term (n=4):
-6 = a₁ + (4-1)d (Equation 1)
For the eleventh term (n=11):
-34 = a₁ + (11-1)d (Equation 2)
Step 2: Simplify the equations:
Equation 1: -6 = a₁ + 3d
Equation 2: -34 = a₁ + 10d
Step 3: Solve the two equations simultaneously to find the values of a₁ and d. We can do this by subtracting Equation 1 from Equation 2:
Equation 2 - Equation 1:
-34 - (-6) = a₁ + 10d - (a₁ + 3d)
-34 + 6 = 7d
-28 = 7d
Divide both sides by 7:
d = -4
Step 4: Substitute the value of d into Equation 1 to find a₁:
-6 = a₁ + 3d
a₁ = -6 - 3(-4)
a₁ = -6 + 12
a₁ = 6
The first term (a₁) is 6, and the common difference (d) is -4. Therefore, the arithmetic sequence is:
6, 2, -2, -6, -10, -14, -18, -22, -26, -30, -34
The (n+1)st term is a+nd
a+3d=-6
a+10d = -34
subtract to get 7d = -28
having d, now you can get a, since
a+3d = -6
a+3d=_6
To find the arithmetic sequence, we need to determine the common difference (d) between consecutive terms. Once we have the common difference, we can determine the entire sequence.
Let's use the information given:
The fourth term, denoted as a₄, is -6.
The eleventh term, denoted as a₁₁, is -34.
We can use the formula for the n-th term of an arithmetic sequence:
aₙ = a₁ + (n - 1) * d
Where aₙ is the n-th term, a₁ is the first term, n is the position of the term in the sequence, and d is the common difference.
We have two equations:
a₄ = a₁ + (4 - 1) * d ---> Equation 1
a₁₁ = a₁ + (11 - 1) * d ---> Equation 2
Substitute the given values:
-6 = a₁ + (4 - 1) * d ---> Equation 1
-34 = a₁ + (11 - 1) * d ---> Equation 2
Simplify the equations:
-6 = a₁ + 3d ---> Equation 1
-34 = a₁ + 10d ---> Equation 2
Now, we have a system of two equations with two variables (a₁ and d). We can solve this system to find the values of a₁ and d.
Subtract Equation 1 from Equation 2 to eliminate a₁:
-34 - (-6) = (a₁ + 10d) - (a₁ + 3d)
-34 + 6 = a₁ + 10d - a₁ - 3d
-28 = 7d
Divide both sides by 7:
-28/7 = d
-4 = d
Now substitute the value of d back into Equation 1 or Equation 2 to find a₁:
-6 = a₁ + 3(-4)
-6 = a₁ - 12
a₁ = -6 + 12
a₁ = 6
We have found that the first term, a₁, is 6 and the common difference, d, is -4.
Therefore, the arithmetic sequence is: 6, 2, -2, -6, -10, -14, -18, -22, -26, -30, -34.