The fourth term of an arithmetic sequence is less than the fifth term by 3. The seventh term is three times the fifth term. Find the common difference and the first term
just translate and use your definitions of an AS
a+3d = a+4d - 3
d = 3, hey, we got the common difference already
a+6d = 3(a+4d)
a + 6d = 3a + 12d
-2a = 6d = 18
a = -9
check:
terms are:
-9 -6 -3 0 3 6 9 12 15 ....
all conditions check out.
20
Let's denote the first term of the arithmetic sequence as "a" and the common difference as "d".
We know that the fourth term is less than the fifth term by 3. In terms of the variables, this can be written as:
a + 3d = a + 4d - 3
Simplifying this equation, we get:
-d = -3
Solving for "d", we find that the common difference is 3.
Next, we are given that the seventh term is three times the fifth term. Using the formula for the nth term of an arithmetic sequence, we can write this as:
a + 6d = 3(a + 4d)
Simplifying this equation, we get:
a + 6d = 3a + 12d
Rearranging terms, we have:
-2a = 6d
Substituting the value of "d" we found earlier, we get:
-2a = 6(3)
Simplifying, we find:
-2a = 18
Dividing both sides by -2, we find:
a = -9
Therefore, the common difference is 3 and the first term is -9.
To find the common difference and the first term of an arithmetic sequence, we can follow these steps:
Step 1: Assign variables
Let's represent the common difference by "d" and the first term by "a."
Step 2: Formulate the equations
Using the given information:
The fourth term is less than the fifth term by 3: a + 3d = a + 4d - 3 ...(Equation 1)
The seventh term is three times the fifth term: a + 6d = 3(a + 4d) ...(Equation 2)
Step 3: Solve the equations
Now, let's solve the system of equations to determine the values of "a" and "d."
Let's start with Equation 1:
a + 3d = a + 4d - 3
Simplifying the equation, we get:
3d = 4d - 3
Subtracting 4d from both sides:
3d - 4d = -3
-d = -3
d = 3
Now that we have the value of "d," we can substitute it into Equation 1 or Equation 2 to find "a." Let's use Equation 1:
a + 3d = a + 4d - 3
Substituting the value of "d" as 3:
a + 3(3) = a + 4(3) - 3
a + 9 = a + 12 - 3
Simplifying the equation, we get:
a + 9 = a + 9
Subtracting "a" from both sides, we get:
9 = 9
Since the variables cancel out and the equation is true, this means that "a" can be any real number.
So, the common difference (d) is 3, and the first term (a) can be any real number.