What is a formula for the nth term of the given sequence?
50, comma, minus, 20, comma, 8, point, point, .
50,−20,8...
Answer
Multiple Choice Answers
a, start subscript, n, end subscript, equals, minus, 125, left parenthesis, minus, start fraction, 2, divided by, 5, end fraction, right parenthesis, start superscript, n, minus, 1, end superscripta 
n
​
 =−125(− 
5
2
​
 ) 
n−1
 
a, start subscript, n, end subscript, equals, minus, 125, left parenthesis, minus, start fraction, 2, divided by, 5, end fraction, right parenthesis, to the power na 
n
​
 =−125(− 
5
2
​
 ) 
n
 
a, start subscript, n, end subscript, equals, 50, left parenthesis, minus, start fraction, 5, divided by, 2, end fraction, right parenthesis, to the power na 
n
​
 =50(− 
2
5
​
 ) 
n
 
a, start subscript, n, end subscript, equals, 50, left parenthesis, minus, start fraction, 2, divided by, 5, end fraction, right parenthesis, to the power na 
n
​
 =50(− 
5
2
​
 ) 
n

To find the formula for the nth term of the given sequence \( 50, -20, 8, \dots \), we will first check if it's a geometric sequence by examining the ratios between consecutive terms.

1. The ratio between the second term and the first term is:
   \[
   r = \frac{-20}{50} = -\frac{2}{5}
   \]

2. The ratio between the third term and the second term is:
   \[
   r = \frac{8}{-20} = -\frac{2}{5}
   \]

Since the ratio is constant, the sequence is geometric with a common ratio of \( -\frac{2}{5} \).

The general formula for the nth term of a geometric sequence is:
\[
a_n = a_1 \cdot r^{n-1}
\]
where \( a_1 \) is the first term and \( r \) is the common ratio.

Here, \( a_1 = 50 \) and \( r = -\frac{2}{5} \). Thus, the formula becomes:
\[
a_n = 50 \cdot \left( -\frac{2}{5} \right)^{n-1}
\]

Now, let's compare this to the multiple choice options:

- The correct formula is:
  \[
  \boxed{a_n = 50 \cdot \left( -\frac{2}{5} \right)^{n-1}}
  \]

Thus, the correct choice is the last option:  
\[
a_n = 50 \left( -\frac{2}{5} \right)^n
\]

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Would you like a more detailed breakdown or have any questions?

Here are 5 related questions to explore:
1. How do you find the sum of an infinite geometric series?
2. What is the difference between an arithmetic and geometric sequence?
3. How do you determine if a sequence is geometric?
4. Can the common ratio in a geometric sequence be zero?
5. How do you calculate the nth term of an arithmetic sequence?

**Tip:** In geometric sequences, if the common ratio's absolute value is less than 1, the terms decrease in magnitude, approaching zero.

What is a formula for the nth term of the given sequence? 50,−20,8...

Learn how to derive the nth term formula for the geometric sequence 50, -20, 8,... This problem demonstrates key concepts such as the geometric sequence formula, common ratios, and calculating the nth term. Suitable for high school students, Grades 9-12.

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