3.4 Is -72 a term in the sequence? Justify your answer.
Which term in the sequence has a value of 8 ?

Ask by Goodwin Watson. in South Africa

Dec 16,2024

Question

3.4 Is -72 a term in the sequence? Justify your answer. Which term in the sequence $$ -2 ;-\frac{5}{3} ;-\frac{4}{3} ;-1 ; \ldots $$ has a value of 8 ?

UpStudy AI · Accepted Answer

To determine if -72 is a term in the given sequence, we first need to identify the pattern of the sequence. Let's look at the differences between consecutive terms:

$$
\begin{align*}
-\frac{5}{3} - (-2) &= -\frac{5}{3} + 2 = -\frac{5}{3} + \frac{6}{3} = \frac{1}{3} \
-\frac{4}{3} - (-\frac{5}{3}) &= -\frac{4}{3} + \frac{5}{3} = \frac{1}{3} \
-1 - (-\frac{4}{3}) &= -1 + \frac{4}{3} = \frac{1}{3}
\end{align*}
$$

It appears that the difference between consecutive terms is consistently $$\frac{1}{3}$$. This suggests that the sequence is an arithmetic sequence with a common difference of $$\frac{1}{3}$$.

To find the term that equals 8, we can use the formula for the nth term of an arithmetic sequence:

$$
a_n = a_1 + (n - 1)d
$$

where $$a_n$$ is the nth term, $$a_1$$ is the first term, $$n$$ is the term number, and $$d$$ is the common difference.

In this case, $$a_1 = -2$$ and $$d = \frac{1}{3}$$. We want to find the value of $$n$$ such that $$a_n = 8$$:

$$
8 = -2 + (n - 1)\frac{1}{3}
$$

Let's solve for $$n$$:

$$
\begin{align*}
8 + 2 &= (n - 1)\frac{1}{3} \
10 &= (n - 1)\frac{1}{3} \
10 \cdot 3 &= n - 1 \
30 &= n - 1 \
n &= 31
\end{align*}
$$

So, the 31st term of the sequence is 8.

Now, let's check if -72 is a term in the sequence. We can use the same formula to find the term number for -72:

$$
-72 = -2 + (n - 1)\frac{1}{3}
$$

Solving for $$n$$:

$$
\begin{align*}
-72 + 2 &= (n - 1)\frac{1}{3} \
-70 &= (n - 1)\frac{1}{3} \
-70 \cdot 3 &= n - 1 \
-210 &= n - 1 \
n &= -209
\end{align*}
$$

Since $$n$$ is a negative number, it is not possible for the sequence to have a negative term number. Therefore, -72 is not a term in the given sequence.
The 31st term of the sequence is 8. -72 is not a term in the sequence.

3.4 Is -72 a term in the sequence? Justify your answer.
Which term in the sequence has a value of 8 ?

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3.4 Is -72 a term in the sequence? Justify your answer. Which term in the sequence has a value of 8 ?