Question
the first five terms of a sequence are
a1=47,a2=33,a3=19,a4=5,a5=-9
based on this information, create an equation that can be used to find the nth term of the sequence, an
move the correct answer to each box. each answer may be used more than once. not all answers will be used.
an= ___n+___
a 61
b 47
c -14
d -61
e 14
a1=47,a2=33,a3=19,a4=5,a5=-9
based on this information, create an equation that can be used to find the nth term of the sequence, an
move the correct answer to each box. each answer may be used more than once. not all answers will be used.
an= ___n+___
a 61
b 47
c -14
d -61
e 14
Answers
GPT-4o mini
To find the nth term of the sequence given the first five terms \( a_1 = 47 \), \( a_2 = 33 \), \( a_3 = 19 \), \( a_4 = 5 \), and \( a_5 = -9 \), we first need to identify the pattern in the sequence.
1. **Calculate the differences between consecutive terms:**
- \( a_2 - a_1 = 33 - 47 = -14 \)
- \( a_3 - a_2 = 19 - 33 = -14 \)
- \( a_4 - a_3 = 5 - 19 = -14 \)
- \( a_5 - a_4 = -9 - 5 = -14 \)
The common difference is \(-14\). This indicates that the sequence is arithmetic with:
- First term \( a = 47 \)
- Common difference \( d = -14 \)
2. **Determine the formula for the nth term of an arithmetic sequence:**
The formula for the nth term \( a_n \) of an arithmetic sequence can be expressed as:
\[
a_n = a + (n - 1) \cdot d
\]
Substituting the values we have:
\[
a_n = 47 + (n - 1)(-14)
\]
Simplifying this gives:
\[
a_n = 47 - 14(n - 1)
\]
\[
= 47 - 14n + 14
\]
\[
= 61 - 14n
\]
Thus, the final equation for the nth term is:
\[
a_n = -14n + 61
\]
3. **Fill in the blanks using the provided options:**
We can match this with the form \( a_n = \_\_n + \_\_ \).
- The coefficient of \( n \) is \(-14\) which can be matched with option \( c \) if you consider \( -14 \) as the value that fits,
- The constant term is \( 61 \) which fits as well and should be matched with \( 61 \).
Thus the filled form is:
\[
a_n = -14n + 61
\]
So we can write down:
- The answer for the coefficient of \( n \) (\(-14\)): **(c)** (as \(-14\))
- The answer for the constant term (61): **(a)**
So, the filled answers are:
\[
a_n = \boxed{c} n + \boxed{a}
\]
Where \( c = -14 \) and \( a = 61 \).
1. **Calculate the differences between consecutive terms:**
- \( a_2 - a_1 = 33 - 47 = -14 \)
- \( a_3 - a_2 = 19 - 33 = -14 \)
- \( a_4 - a_3 = 5 - 19 = -14 \)
- \( a_5 - a_4 = -9 - 5 = -14 \)
The common difference is \(-14\). This indicates that the sequence is arithmetic with:
- First term \( a = 47 \)
- Common difference \( d = -14 \)
2. **Determine the formula for the nth term of an arithmetic sequence:**
The formula for the nth term \( a_n \) of an arithmetic sequence can be expressed as:
\[
a_n = a + (n - 1) \cdot d
\]
Substituting the values we have:
\[
a_n = 47 + (n - 1)(-14)
\]
Simplifying this gives:
\[
a_n = 47 - 14(n - 1)
\]
\[
= 47 - 14n + 14
\]
\[
= 61 - 14n
\]
Thus, the final equation for the nth term is:
\[
a_n = -14n + 61
\]
3. **Fill in the blanks using the provided options:**
We can match this with the form \( a_n = \_\_n + \_\_ \).
- The coefficient of \( n \) is \(-14\) which can be matched with option \( c \) if you consider \( -14 \) as the value that fits,
- The constant term is \( 61 \) which fits as well and should be matched with \( 61 \).
Thus the filled form is:
\[
a_n = -14n + 61
\]
So we can write down:
- The answer for the coefficient of \( n \) (\(-14\)): **(c)** (as \(-14\))
- The answer for the constant term (61): **(a)**
So, the filled answers are:
\[
a_n = \boxed{c} n + \boxed{a}
\]
Where \( c = -14 \) and \( a = 61 \).