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Magazine Chapter 10: Problem 16
Find the sum, if it exists. $$31500+6300+1260+252+\cdots$$
Short Answer
Expert verified
The sum of the series is 39375.
Step by step solution
01
Identify the series
The series given is \( 31500 + 6300 + 1260 + 252 + \cdots \). This looks like a geometric series because each term after the first is obtained by multiplying the previous term by a common ratio.
02
Find the common ratio
To find the common ratio, divide the second term by the first term: \( \frac{6300}{31500} = \frac{1}{5} \). Now check with the next term: \( \frac{1260}{6300} = \frac{1}{5} \). So, the common ratio \( r = \frac{1}{5} \).
03
Check if the series converges
For an infinite geometric series to have a sum, the common ratio \( r \) must satisfy \( |r| < 1 \). Here, \( |\frac{1}{5}| = 0.2 < 1 \). Thus, the series converges.
04
Apply the formula for the sum of a convergent geometric series
The sum \( S \) of an infinite convergent geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = 31500 \) and \( r = \frac{1}{5} \).
05
Calculate the sum
Substitute the values into the formula to find the sum: \[ S = \frac{31500}{1 - \frac{1}{5}} = \frac{31500}{\frac{4}{5}} = 31500 \times \frac{5}{4} \] Calculate: \[ S = 31500 \times 1.25 = 39375 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Ratio
In a geometric series, the common ratio is the factor by which you multiply each term to get the next term. It's a key characteristic that defines the series as geometric.
- To find the common ratio in a sequence, divide any term by the previous term. For example, in the series given, divide 6300 by 31500 which gives you \( \frac{1}{5} \).
- It's essential to ensure that this ratio is consistent across other terms. So, dividing 1260 by 6300 should also yield \( \frac{1}{5} \).
- If \( |r| > 1 \), the series grows larger.
- If \( |r| < 1 \), the series diminishes, indicating that it might converge or "settle down" to a specific sum.
Convergence
Convergence in the context of an infinite series determines whether the series approaches a specific value as more terms are added. For a geometric series, convergence depends on the common ratio, \( r \).
In the example series, the common ratio \( r = \frac{1}{5} \) has an absolute value of 0.2, which is less than 1, confirming that the series will converge. This is crucial because only a convergent series has a calculable sum of its infinite terms based on a known formula.
- A geometric series converges when \( |r| < 1 \).
- This means the absolute value of the common ratio needs to be less than 1.
In the example series, the common ratio \( r = \frac{1}{5} \) has an absolute value of 0.2, which is less than 1, confirming that the series will converge. This is crucial because only a convergent series has a calculable sum of its infinite terms based on a known formula.
Sum of Infinite Series
Once it's established that a geometric series converges, we can calculate the sum of its infinite number of terms using the sum formula:\[ S = \frac{a}{1 - r} \]where \( a \) is the first term of the series and \( r \) is the common ratio.
- The presence of \( 1 - r \) in the denominator signifies that the series would not have a finite sum if \( |r| \) were equal to or greater than 1.
- This formula effectively compresses all the contributions of the infinite terms into a single finite number.
- First calculate \( 1 - r = 1 - \frac{1}{5} = \frac{4}{5} \).
- Then calculate the sum \( S = \frac{31500}{\frac{4}{5}} = 31500 \times \frac{5}{4} \).
- Finally, multiplying gives \( S = 39375 \).
