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Chapter 5: Problem 29

Find the twentieth term and sum of the first 20 terms of the geometric progression \(-3,3,-3,3, \ldots\)

Short Answer

Expert verified
The twentieth term (T20) in the geometric progression is 3, and the sum of the first 20 terms (S20) is 0.

Step by step solution

01

Find the common ratio (r)

The common ratio (r) is the ratio of any term to its preceding term. In this case, we are given the first few terms of the sequence: -3, 3, -3, 3, ... . To find the common ratio, divide the second term by the first term: \( r = \frac{3}{-3} = -1 \).
02

Find the general term formula

The general term formula for a geometric progression is given by: \( T_n = ar^{n-1} \), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. In this case, the first term (a) is -3 and the common ratio (r) is -1.
03

Calculate the twentieth term (T20)

Now we will substitute the values we found into the general term formula to calculate the twentieth term, T20: \[T_{20} = ar^{20-1} = (-3)(-1)^{19}\] Since -1 raised to an odd power is -1, this simplifies to: \[T_{20} = (-3)(-1) = 3\] So, the twentieth term (T20) is 3.
04

Calculate the sum of the first 20 terms (S20)

To find the sum of the first \(n\) terms of a geometric progression, we use the formula: \[ S_n = \frac{a(r^n - 1)}{r - 1} \] Substitute the values we found (a = -3, r = -1, and n = 20) into the formula: \[ S_{20} = \frac{-3((-1)^{20} - 1)}{-1 - 1} \] Since -1 raised to an even power is 1, this simplifies to: \[ S_{20} = \frac{-3(1 - 1)}{-2} = \frac{-3(0)}{-2} = 0 \] Thus, the sum of the first 20 terms (S20) is 0. In conclusion: The twentieth term (T20) in the geometric progression is 3, and the sum of the first 20 terms (S20) is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the *common ratio*. To determine the common ratio in a sequence, one can simply take any term and divide it by its preceding term. For example, in the series
  • -3,
  • 3,
  • -3,
  • 3,
we have a first term of -3 and a second term of 3. Thus, the common ratio can be computed as \( r = \frac{3}{-3} = -1 \). This means each term is obtained by multiplying the previous term by -1. In geometric sequences, understanding and identifying the common ratio is essential, as it helps predict future terms and understand the sequence's behavior.
General Term Formula
In any geometric progression, the general term formula is an equation that allows us to find any term in the sequence without listing all the previous terms. This formula is expressed as:\[ T_n = ar^{n-1} \]where:
  • \( T_n \) is the \( n^{th} \) term we want to find,
  • \( a \) is the first term of the sequence,
  • \( r \) is the common ratio,
  • \( n \) is the term number.
For the sequence given in our problem, \( a = -3 \) and \( r = -1 \). To find the twentieth term (\( T_{20} \)), we substitute the values into our formula: \[ T_{20} = (-3)(-1)^{20-1} = (-3)(-1)^{19} = 3 \]This shows that the 20th term in the sequence is 3. This powerful formula is handy because it allows us to compute the 20th term, the 100th term, or any other without listing each term.
Sum of Geometric Series
The sum of a geometric series is calculated when we want to find the total of the first \( n \) terms of a geometric progression. For this, we use the sum formula:\[ S_n = \frac{a(r^n - 1)}{r - 1} \]where:
  • \( S_n \) is the sum of the first \( n \) terms,
  • \( a \) is the first term,
  • \( r \) is the common ratio,
  • \( n \) is the number of terms to sum.
Applying this to our series, with \( a = -3 \), \( r = -1 \), and \( n = 20 \), the sum calculation goes as follows:\[ S_{20} = \frac{-3((-1)^{20} - 1)}{-1 - 1} = \frac{-3(1 - 1)}{-2} = \frac{-3(0)}{-2} = 0 \]Thus, the sum of the first 20 terms turns out to be 0. This outcome occurs because the alternating pattern of positive and negative terms cancels each other out, a common occurrence in geometric progressions with a negative common ratio.

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