To determine which term in the sequence [tex]\( T_n = n(n + 2) \)[/tex] is equal to 143, we need to solve the equation [tex]\( n(n + 2) = 143 \)[/tex].
Let's proceed step-by-step:
1. Setting up the equation:
We start with the given sequence formula:
[tex]\[
T_n = n(n + 2)
\][/tex]
and set it equal to 143:
[tex]\[
n(n + 2) = 143
\][/tex]
2. Forming a quadratic equation:
Expand the left-hand side:
[tex]\[
n^2 + 2n = 143
\][/tex]
Then, rearrange the equation to standard quadratic form:
[tex]\[
n^2 + 2n - 143 = 0
\][/tex]
3. Solving the quadratic equation:
To solve the quadratic equation [tex]\( n^2 + 2n - 143 = 0 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -143 \)[/tex]. Plugging in these values, we have:
[tex]\[
n = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-143)}}{2 \cdot 1}
\][/tex]
Simplifying inside the square root:
[tex]\[
n = \frac{-2 \pm \sqrt{4 + 572}}{2}
\][/tex]
[tex]\[
n = \frac{-2 \pm \sqrt{576}}{2}
\][/tex]
Since [tex]\( \sqrt{576} = 24 \)[/tex], the equation becomes:
[tex]\[
n = \frac{-2 \pm 24}{2}
\][/tex]
4. Finding the roots:
This gives us two potential solutions:
[tex]\[
n = \frac{-2 + 24}{2} = \frac{22}{2} = 11
\][/tex]
and
[tex]\[
n = \frac{-2 - 24}{2} = \frac{-26}{2} = -13
\][/tex]
Therefore, the solutions to the equation are [tex]\( n = 11 \)[/tex] and [tex]\( n = -13 \)[/tex].
Since [tex]\( n = -13 \)[/tex] is a negative term and may not typically be considered in the context of sequence terms, we focus on the positive integer solution.
Conclusion:
The term in the sequence [tex]\( T_n = n(n + 2) \)[/tex] that equals 143 is the 11th term.
