Determine the arithmetic sequence
if the fourth term is -6 and the
eleventh term is -34.

Question

Determine the arithmetic sequence if the fourth term is -6 and the eleventh term is -34.

UpStudy AI · Accepted Answer

To determine the arithmetic sequence, we need to find the common difference (d) and the first term (a₁) of the sequence.

Given:
- The fourth term (a₄) is -6.
- The eleventh term (a₁₁) is -34.

The formula for the nth term of an arithmetic sequence is:
$$ a_n = a_1 + (n-1)d $$

We can use the given information to set up two equations:
1. For the fourth term:
$$ a_4 = a_1 + 3d = -6 $$
2. For the eleventh term:
$$ a_{11} = a_1 + 10d = -34 $$

Now, we can solve these two equations to find the values of $$ a_1 $$ and $$ d $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}a_{1}+3d=-6\a_{1}+10d=-34\end{array}\right.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}a_{1}=-6-3d\a_{1}+10d=-34\end{array}\right.$$
- step2: Substitute the value of $$a_{1}:$$
  $$-6-3d+10d=-34$$
- step3: Add the terms:
  $$-6+7d=-34$$
- step4: Move the constant to the right side:
  $$7d=-34+6$$
- step5: Add the numbers:
  $$7d=-28$$
- step6: Divide both sides:
  $$\frac{7d}{7}=\frac{-28}{7}$$
- step7: Divide the numbers:
  $$d=-4$$
- step8: Substitute the value of $$d:$$
  $$a_{1}=-6-3\left(-4\right)$$
- step9: Simplify:
  $$a_{1}=6$$
- step10: Calculate:
  $$\left\{ \begin{array}{l}a_{1}=6\d=-4\end{array}\right.$$
- step11: Check the solution:
  $$\left\{ \begin{array}{l}a_{1}=6\d=-4\end{array}\right.$$
- step12: Rewrite:
  $$\left(a_{1},d\right) = \left(6,-4\right)$$
The solution to the system of equations is:
- $$ a_1 = 6 $$
- $$ d = -4 $$

Therefore, the arithmetic sequence is:
$$ 6, 2, -2, -6, -10, -14, -18, -22, -26, -30, -34, \ldots $$
The arithmetic sequence is 6, 2, -2, -6, -10, -14, -18, -22, -26, -30, -34, ...

Determine the arithmetic sequence
if the fourth term is -6 and the
eleventh term is -34.

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