We appreciate your visit to Which of the following statements is true about the sequence below 1 6 36 216 1 296 7 776 A The sequence is arithmetic B. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
We are given the sequence
[tex]$$
1,\ 6,\ 36,\ 216,\ 1296,\ 7776,\ \dots
$$[/tex]
and several statements about it. Let’s check each statement step by step.
1. To decide whether the sequence is geometric or arithmetic, we begin by looking at the ratio of successive terms. Calculate the ratio between the second and first terms:
[tex]$$
\frac{6}{1} = 6.
$$[/tex]
Next, calculate the ratio for the other pairs:
[tex]$$
\frac{36}{6} = 6,\quad \frac{216}{36} = 6,\quad \frac{1296}{216} = 6,\quad \frac{7776}{1296} = 6.
$$[/tex]
Because the common ratio is constant and equal to [tex]$6$[/tex], the sequence is geometric (not arithmetic).
2. Since the sequence is geometric, the general term is given by
[tex]$$
a_n = a_1 \cdot r^{\,n-1}.
$$[/tex]
Here, [tex]$a_1 = 1$[/tex] and [tex]$r=6$[/tex]. This confirms that the common ratio is indeed [tex]$6$[/tex], making the statement “The common ratio is 6” true.
3. Now, compute the next term in the sequence (the term after [tex]$7776$[/tex]). Multiply the last given term by the common ratio:
[tex]$$
7776 \times 6 = 46656.
$$[/tex]
Thus, “The next term in the sequence is 46,656” is also true.
4. For the 10th term, we use the formula:
[tex]$$
a_{10} = 1 \cdot 6^{10-1} = 6^9.
$$[/tex]
Evaluating [tex]$6^9$[/tex] gives a value much larger than [tex]$10,000$[/tex]. Therefore, the statement “The 10th term in the sequence is 10,000” is false.
5. Lastly, regarding the arithmetic nature: The sequence would be arithmetic if the difference between successive terms were constant. However, the differences are:
[tex]$$
6-1 = 5,\quad 36-6 = 30,\quad 216-36 = 180,\quad \dots
$$[/tex]
Since the differences are not constant, the sequence is not arithmetic, and any statement claiming a common difference (such as “The common difference is 6”) is false.
In summary, the true statements are:
- “The common ratio is 6.”
- “The next term in the sequence is 46,656.”
- “The sequence is geometric.”
Thus, the true statements are numbers 2, 4, and 5.
[tex]$$
1,\ 6,\ 36,\ 216,\ 1296,\ 7776,\ \dots
$$[/tex]
and several statements about it. Let’s check each statement step by step.
1. To decide whether the sequence is geometric or arithmetic, we begin by looking at the ratio of successive terms. Calculate the ratio between the second and first terms:
[tex]$$
\frac{6}{1} = 6.
$$[/tex]
Next, calculate the ratio for the other pairs:
[tex]$$
\frac{36}{6} = 6,\quad \frac{216}{36} = 6,\quad \frac{1296}{216} = 6,\quad \frac{7776}{1296} = 6.
$$[/tex]
Because the common ratio is constant and equal to [tex]$6$[/tex], the sequence is geometric (not arithmetic).
2. Since the sequence is geometric, the general term is given by
[tex]$$
a_n = a_1 \cdot r^{\,n-1}.
$$[/tex]
Here, [tex]$a_1 = 1$[/tex] and [tex]$r=6$[/tex]. This confirms that the common ratio is indeed [tex]$6$[/tex], making the statement “The common ratio is 6” true.
3. Now, compute the next term in the sequence (the term after [tex]$7776$[/tex]). Multiply the last given term by the common ratio:
[tex]$$
7776 \times 6 = 46656.
$$[/tex]
Thus, “The next term in the sequence is 46,656” is also true.
4. For the 10th term, we use the formula:
[tex]$$
a_{10} = 1 \cdot 6^{10-1} = 6^9.
$$[/tex]
Evaluating [tex]$6^9$[/tex] gives a value much larger than [tex]$10,000$[/tex]. Therefore, the statement “The 10th term in the sequence is 10,000” is false.
5. Lastly, regarding the arithmetic nature: The sequence would be arithmetic if the difference between successive terms were constant. However, the differences are:
[tex]$$
6-1 = 5,\quad 36-6 = 30,\quad 216-36 = 180,\quad \dots
$$[/tex]
Since the differences are not constant, the sequence is not arithmetic, and any statement claiming a common difference (such as “The common difference is 6”) is false.
In summary, the true statements are:
- “The common ratio is 6.”
- “The next term in the sequence is 46,656.”
- “The sequence is geometric.”
Thus, the true statements are numbers 2, 4, and 5.
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