The fifth term of an arithmetic sequence is equal to 6 and the sum of the first 12 terms is equal to 45. Find the first term and the common difference.

having trouble with this, can't figure out the steps?

Well, it seems like this arithmetic sequence is playing hard to get! Let's solve this puzzle together.

First, let's find the common difference, denoted by 'd'. We can use the formula for the nth term of an arithmetic sequence:

An = A1 + (n-1) * d

Given that the fifth term (A5) is 6, we can substitute the values into the equation:

6 = A1 + (5-1) * d
6 = A1 + 4d

Now, let's work our magic to find the sum of the first 12 terms using the formula for the sum of an arithmetic series. It goes like this:

Sn = (n/2) * (2A1 + (n-1) * d)

Substituting the given values of Sn (45) and n (12) into the equation:

45 = (12/2) * (2A1 + (12-1) * d)
45 = 6 * (2A1 + 11d)

Now, my friend, we have two equations with two unknowns (A1 and d). We can solve this system of equations using a method called substitution or elimination. Let's get down to business and use substitution.

From the first equation, we can isolate A1:

A1 = 6 - 4d

Now, substitute A1 into the second equation:

45 = 6 * (2(6 - 4d) + 11d)
45 = 6 * (12 - 8d + 11d)
45 = 6 * (12 + 3d)
45 = 72 + 18d
-27 = 18d
d = -27/18
d = -3/2

Great! We've got the value for the common difference (d). Now, let's plug it back into the first equation to find the first term (A1):

6 = A1 + 4(-3/2)
6 = A1 - 6
A1 = 12

Congratulations! The first term of the arithmetic sequence is 12, and the common difference is -3/2. Time to celebrate, my friend!

To find the first term and the common difference of an arithmetic sequence, we can use the given information about the fifth term and the sum of the first 12 terms.

Let's denote the first term of the sequence as 'a' and the common difference as 'd'.

Step 1: Finding the fifth term:
We are given that the fifth term is equal to 6. The formula to find the nth term of an arithmetic sequence is given by:
nth term = a + (n - 1)d

Substituting the given values, we have:
6 = a + (5 - 1)d
6 = a + 4d
This is our first equation.

Step 2: Finding the sum of the first 12 terms:
We are given that the sum of the first 12 terms is equal to 45. The formula to find the sum of an arithmetic series is given by:
Sum = (n/2)(2a + (n - 1)d)

Substituting the given values, we have:
45 = (12/2)(2a + (12 - 1)d)
45 = 6(2a + 11d)
Simplifying further, we get:
45 = 12a + 66d
This is our second equation.

Step 3: Solving the system of equations:
We will solve the system of equations consisting of our first and second equations. Here, we have two equations with two variables (a and d).

First, let's manipulate the first equation to solve for 'a':
6 = a + 4d
Rearranging, we get:
a = 6 - 4d

Substituting this expression for 'a' into the second equation:
45 = 12a + 66d
45 = 12(6 - 4d) + 66d
45 = 72 - 48d + 66d
45 = 72 + 18d
18d = 45 - 72
18d = -27
d = -27/18
d = -1.5

Step 4: Finding the first term:
Now that we have the value of 'd', we can substitute it back into the first equation to find 'a':
6 = a + 4(-1.5)
6 = a - 6
a = 6 + 6
a = 12

Therefore, the first term (a) of the arithmetic sequence is 12, and the common difference (d) is -1.5.

To find the first term and the common difference of an arithmetic sequence, we can follow these steps:

Step 1: Identify the given information.
- The fifth term of the sequence is equal to 6.
- The sum of the first 12 terms is equal to 45.

Step 2: Write the formula for the nth term of an arithmetic sequence.
The formula for the nth term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n-1)d \]
where:
- \( a_n \) is the nth term
- \( a_1 \) is the first term
- \( d \) is the common difference
- \( n \) is the position of the term we want to find

Step 3: Find the first term.
Since we know the value of the fifth term (\( a_5 \)) is 6, we can substitute those values into the formula:
\[ a_5 = a_1 + 4d = 6 \]
This equation allows us to express \( a_1 \) in terms of \( d \) as follows:
\[ a_1 = 6 - 4d \]

Step 4: Find the sum of the first 12 terms.
We are given that the sum of the first 12 terms is equal to 45. The formula for the sum of an arithmetic sequence is given by:
\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]
Where:
- \( S_n \) is the sum of the first n terms
- \( n \) is the number of terms

We can substitute the given values into the formula:
\[ S_{12} = \frac{12}{2} (2a_1 + 11d) = 45 \]

Step 5: Solve the equations.
Now we have two equations that involve \( a_1 \) and \( d \):
\[ a_1 = 6 - 4d \]
\[ 6 (2a_1 + 11d) = 45 \]
Solving these equations simultaneously will give us the values of \( a_1 \) and \( d \).

Substituting the value of \( a_1 \) in the second equation:
\[ 6(2(6-4d) + 11d) = 45 \]
\[ 6(12 - 8d + 11d) = 45 \]
\[ 72 + 18d = 45 \]
\[ 18d = 45 - 72 \]
\[ 18d = -27 \]
\[ d = -1.5 \]

Substituting the value of \( d \) in the equation for \( a_1 \):
\[ a_1 = 6 - 4(-1.5) \]
\[ a_1 = 6 + 6 \]
\[ a_1 = 12 \]

Therefore, the first term of the arithmetic sequence is 12 and the common difference is -1.5.

f = first term , d = difference

f + 4d = 6 ... 12f + 48d = 72

12f + 66d = 45

18d = -27