Solved on Jan 18, 2024

Find the first term $u_1$ and the common difference $d$ given $u_7=11$ and $u_{12}=56$ in an arithmetic sequence.

STEP 1

Assumptions
1. The sequence is arithmetic, meaning there is a common difference $d$ between each term.
2. $u_7$ represents the 7th term in the sequence, which is 11.
3. $u_{12}$ represents the 12th term in the sequence, which is 56.
4. $u_1$ represents the first term in the sequence, which we need to find.
5. $d$ represents the common difference between the terms, which we also need to find.

STEP 2

The formula for the $n$ -th term of an arithmetic sequence is given by:
$u_n = u_1 + (n - 1)d$

STEP 3

We can set up two equations using the given terms $u_7$ and $u_{12}$ :
$u_7 = u_1 + 6d$ $u_{12} = u_1 + 11d$

STEP 4

Substitute the given values for $u_7$ and $u_{12}$ into the equations:
$11 = u_1 + 6d$ $56 = u_1 + 11d$

STEP 5

We now have a system of two equations with two unknowns:
\begin{align} u_1 + 6d &= 11 \\ u_1 + 11d &= 56 \end{align}

STEP 6

Subtract the first equation from the second equation to eliminate $u_1$ and solve for $d$ :
$(u_1 + 11d) - (u_1 + 6d) = 56 - 11$

STEP 7

Simplify the equation to find $d$ :
$11d - 6d = 56 - 11$

STEP 8

Combine like terms:
$5d = 45$

STEP 9

Divide both sides by 5 to solve for $d$ :
$d = \frac{45}{5}$

STEP 10

Calculate the value of $d$ :
$d = 9$

STEP 11

Now that we have the value of $d$ , we can substitute it back into one of the original equations to find $u_1$ . Let's use the first equation:
$u_1 + 6d = 11$

STEP 12

Substitute the value of $d$ into the equation:
$u_1 + 6(9) = 11$

STEP 13

Simplify the equation:
$u_1 + 54 = 11$

STEP 14

Subtract 54 from both sides to solve for $u_1$ :
$u_1 = 11 - 54$

STEP 15

Calculate the value of $u_1$ :
$u_1 = -43$
The first term of the sequence is $u_1 = -43$ and the common difference is $d = 9$ .

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Find the first term $u_1$ and the common difference $d$ given $u_7=11$ and $u_{12}=56$ in an arithmetic sequence.

STEP 1

STEP 2

The formula for the $n$ -th term of an arithmetic sequence is given by:
$u_n = u_1 + (n - 1)d$

STEP 3

We can set up two equations using the given terms $u_7$ and $u_{12}$ :
$u_7 = u_1 + 6d$ $u_{12} = u_1 + 11d$

STEP 4

Substitute the given values for $u_7$ and $u_{12}$ into the equations:
$11 = u_1 + 6d$ $56 = u_1 + 11d$

STEP 5

We now have a system of two equations with two unknowns:
\begin{align} u_1 + 6d &= 11 \\ u_1 + 11d &= 56 \end{align}

STEP 6

Subtract the first equation from the second equation to eliminate $u_1$ and solve for $d$ :
$(u_1 + 11d) - (u_1 + 6d) = 56 - 11$

STEP 7

Simplify the equation to find $d$ :
$11d - 6d = 56 - 11$

STEP 8

Combine like terms:
$5d = 45$

STEP 9

Divide both sides by 5 to solve for $d$ :
$d = \frac{45}{5}$

STEP 10

Calculate the value of $d$ :
$d = 9$

STEP 11

Now that we have the value of $d$ , we can substitute it back into one of the original equations to find $u_1$ . Let's use the first equation:
$u_1 + 6d = 11$

STEP 12

Substitute the value of $d$ into the equation:
$u_1 + 6(9) = 11$

STEP 13

Simplify the equation:
$u_1 + 54 = 11$

STEP 14

Subtract 54 from both sides to solve for $u_1$ :
$u_1 = 11 - 54$

STEP 15

Calculate the value of $u_1$ :
$u_1 = -43$
The first term of the sequence is $u_1 = -43$ and the common difference is $d = 9$ .

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Find the first term u1u_1u1​ and the common difference ddd given u7=11u_7=11u7​=11 and u12=56u_{12}=56u12​=56 in an arithmetic sequence.

STEP 1

STEP 2

The formula for the nnn-th term of an arithmetic sequence is given by:un=u1+(n−1)du_n = u_1 + (n - 1)dun​=u1​+(n−1)d

STEP 3

We can set up two equations using the given terms u7u_7u7​ and u12u_{12}u12​:u7=u1+6du_7 = u_1 + 6du7​=u1​+6d u12=u1+11du_{12} = u_1 + 11du12​=u1​+11d

STEP 4

Substitute the given values for u7u_7u7​ and u12u_{12}u12​ into the equations:11=u1+6d11 = u_1 + 6d11=u1​+6d 56=u1+11d56 = u_1 + 11d56=u1​+11d

STEP 5

We now have a system of two equations with two unknowns:\begin{align*} u_1 + 6d &= 11 \\ u_1 + 11d &= 56 \end{align*}

STEP 6

Subtract the first equation from the second equation to eliminate u1u_1u1​ and solve for ddd:(u1+11d)−(u1+6d)=56−11(u_1 + 11d) - (u_1 + 6d) = 56 - 11(u1​+11d)−(u1​+6d)=56−11

STEP 7

Simplify the equation to find ddd:11d−6d=56−1111d - 6d = 56 - 1111d−6d=56−11

STEP 8

Combine like terms:5d=455d = 455d=45

STEP 9

Divide both sides by 5 to solve for ddd:d=455d = \frac{45}{5}d=545​

STEP 10

Calculate the value of ddd:d=9d = 9d=9

STEP 11

Now that we have the value of ddd, we can substitute it back into one of the original equations to find u1u_1u1​. Let's use the first equation:u1+6d=11u_1 + 6d = 11u1​+6d=11

STEP 12

Substitute the value of ddd into the equation:u1+6(9)=11u_1 + 6(9) = 11u1​+6(9)=11

STEP 13

Simplify the equation:u1+54=11u_1 + 54 = 11u1​+54=11

STEP 14

Subtract 54 from both sides to solve for u1u_1u1​:u1=11−54u_1 = 11 - 54u1​=11−54

STEP 15

Calculate the value of u1u_1u1​:u1=−43u_1 = -43u1​=−43The first term of the sequence is u1=−43u_1 = -43u1​=−43 and the common difference is d=9d = 9d=9.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the first term $u_1$ and the common difference $d$ given $u_7=11$ and $u_{12}=56$ in an arithmetic sequence.

The formula for the $n$ -th term of an arithmetic sequence is given by:
$u_n = u_1 + (n - 1)d$

We can set up two equations using the given terms $u_7$ and $u_{12}$ :
$u_7 = u_1 + 6d$ $u_{12} = u_1 + 11d$

Substitute the given values for $u_7$ and $u_{12}$ into the equations:
$11 = u_1 + 6d$ $56 = u_1 + 11d$

We now have a system of two equations with two unknowns:
\begin{align} u_1 + 6d &= 11 \\ u_1 + 11d &= 56 \end{align}

Subtract the first equation from the second equation to eliminate $u_1$ and solve for $d$ :
$(u_1 + 11d) - (u_1 + 6d) = 56 - 11$

Simplify the equation to find $d$ :
$11d - 6d = 56 - 11$

Combine like terms:
$5d = 45$

Divide both sides by 5 to solve for $d$ :
$d = \frac{45}{5}$

Calculate the value of $d$ :
$d = 9$

Now that we have the value of $d$ , we can substitute it back into one of the original equations to find $u_1$ . Let's use the first equation:
$u_1 + 6d = 11$

Substitute the value of $d$ into the equation:
$u_1 + 6(9) = 11$

Simplify the equation:
$u_1 + 54 = 11$

Subtract 54 from both sides to solve for $u_1$ :
$u_1 = 11 - 54$

Calculate the value of $u_1$ :
$u_1 = -43$
The first term of the sequence is $u_1 = -43$ and the common difference is $d = 9$ .