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Magazine Chapter 11: Problem 12
Find \(S_{n}\) for each geometric series described. $$ a_{1}=12, a_{5}=972, r=-3 $$
Short Answer
Expert verified
The sum of the first 5 terms, \(S_5\), is 732.
Step by step solution
01
Determine the formula for the nth term of a geometric sequence
The formula for the nth term in a geometric series is given by:\[ a_n = a_1 imes r^{n-1} \]where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
02
Calculate first term's conditions
We know that \(a_5 = 972\), \(a_1 = 12\), and \(r = -3\). According to the formula for the nth term, substitute these values into the equation for the 5th term:\[ 972 = 12 imes (-3)^{4} \]
03
Solve the equation for the 5th term
Calculate \((-3)^4\) first:\[ (-3)^4 = 81 \]Then multiply by \(a_1\):\[ 12 imes 81 = 972 \]This confirms that our values are correct.
04
Appropriately apply the sum formula for a geometric series
The formula for the sum of the first \(n\) terms of a geometric series is:\[ S_n = a_1 \frac{1-r^n}{1-r} \]This works when \(r eq 1\).
05
Calculate the sum for the first 5 terms
We will calculate \(S_5\) since the problem provides information for the first 5 terms:\[ S_5 = 12 \frac{1 - (-3)^5}{1 - (-3)} \]Calculate \((-3)^5\) first:\[ (-3)^5 = -243 \]Then substitute in the summation formula:\[ S_5 = 12 \frac{1 - (-243)}{1 + 3} \]
06
Simplify the expression to find \(S_5\)
Simplify \(1 - (-243)\):\[ 1 + 243 = 244 \]Then divide by \(1 + 3\):\[ S_5 = 12 \times \frac{244}{4} = 12 \times 61 \]Finally, calculate:\[ S_5 = 732 \]
07
Confirm the result
Review each calculation to ensure correctness at each step. In particular, confirm that all arithmetic operations are correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
nth term formula
One of the key aspects of understanding a geometric sequence is knowing how to find the nth term. The nth term formula is crucial:
- The formula is:\[ a_n = a_1 \times r^{n-1} \]
- \(a_1\) is the first term in the sequence.
- \(r\) is the common ratio.
- \(n\) denotes the term's position in the sequence.
geometric sequence
A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a constant, known as the common ratio.
- A geometric sequence forms a predictable pattern, such as 2, 6, 18, 54, etc.
- Every term is produced by multiplying the previous one by the common ratio \(r\).
- For example, if \(a_1 = 2\) and \(r = 3\), then each term is generated by \(a_n = 2 \times 3^{n-1}\).
- The difference between each term isn't constant, but the ratio is.
common ratio
The common ratio is a critical concept in geometric sequences. It dictates the sequence's growth or decay by determining how each term relates to the previous.
- The common ratio \(r\) is calculated by dividing any term by the term preceding it.
- For instance, in the sequence 3, 12, 48, the common ratio \(r = \frac{12}{3} = 4\).
- \(r\) influences whether the sequence expands (\(r > 1\)) or contracts (\(0 < r < 1\)) or oscillates if \(r\) is a negative number.
- A negative \(r\) causes the sequence to alternate signs.
sum of series formula
Geometric series often require calculating the sum of a specific number of terms, which involves using a unique formula:
- The sum of the first \(n\) terms is calculated by: \[ S_n = a_1 \frac{1-r^n}{1-r} \]
- This formula applies when \(r eq 1\).
- For example, if \(a_1 = 12\), \(r = -3\), and you need \(S_5\): \[ S_5 = 12 \frac{1 - (-3)^5}{1 + 3} \] This simplifies to: \[ S_5 = 12 \times 61 = 732 \]
- This formula helps gather a series of terms into one cohesive sum, showing the total over a defined range.
