Problem 29
Question
Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 14 terms of the geometric sequence: $$-\frac{3}{2}, 3,-6,12, \ldots$$
Step-by-Step Solution
Verified Answer
The sum of the first 14 terms of the geometric sequence is 8193.
1Step 1: Identify the first term and the common ratio
The first term a of the given sequence is -3/2. The common ratio r can be found by dividing the second term by the first term, or the third term by the second term, etc. Here, r= 3 / (-3/2) = -2.
2Step 2: Apply the formula
Now that the first term and the common ratio are known, they can be substituted into the formula for the sum S of the first n terms of a geometric sequence. n is 14 in this exercise. So, \( S = \frac{-3/2 (1 - (-2)^{14})}{1 - (-2)} \).
3Step 3: Simplify the expression
Simplify the equation to find S. The calculation is quite straightforward by using the order of operations in mathematics (parentheses, exponents, multiplication, division, addition, subtraction), leading to \( S = \frac{-3/2 (1 - 16384)}{-3} \). After simplifying, S = 8193.
Key Concepts
Geometric SequenceCommon RatioSum FormulaOrder of Operations
Geometric Sequence
Understanding a geometric sequence is crucial when tackling problems involving repeated multiplication. A geometric sequence, also known as a geometric progression, refers to a series of numbers where each term after the first is found by multiplying the previous term by a fixed number, known as the common ratio.
For instance, in the sequence \( -\frac{3}{2}, 3, -6, 12, \ldots \), each term is generated by multiplying the previous term by -2. Notably, geometric sequences can exhibit growth or decay depending on the magnitude of the common ratio. If the absolute value of the common ratio is greater than one, the sequence grows with each term and if the common ratio is between zero and one, the sequence decreases.
For instance, in the sequence \( -\frac{3}{2}, 3, -6, 12, \ldots \), each term is generated by multiplying the previous term by -2. Notably, geometric sequences can exhibit growth or decay depending on the magnitude of the common ratio. If the absolute value of the common ratio is greater than one, the sequence grows with each term and if the common ratio is between zero and one, the sequence decreases.
Common Ratio
The common ratio is a pivotal aspect of a geometric sequence. It dictates the pattern and direction of the sequence's progression. This ratio is constant for any consecutive terms within the sequence and is obtained by dividing any term by its immediate predecessor.
For the given sequence \( -\frac{3}{2}, 3, -6, 12, \ldots \), the common ratio \( r \) is determined by dividing the second term by the first (\(\frac{3}{-3/2} = -2\)). It is important to ensure the division is made in the correct order to ascertain the correct common ratio. In sequences where terms alternate in sign, the common ratio is negative, contributing to the alternating nature of the sequence.
For the given sequence \( -\frac{3}{2}, 3, -6, 12, \ldots \), the common ratio \( r \) is determined by dividing the second term by the first (\(\frac{3}{-3/2} = -2\)). It is important to ensure the division is made in the correct order to ascertain the correct common ratio. In sequences where terms alternate in sign, the common ratio is negative, contributing to the alternating nature of the sequence.
Sum Formula
Calculating the sum of a geometric sequence can seem daunting, but with the correct formula, it's a breeze. The sum formula is essential for determining the cumulative value of terms in a finite geometric sequence. This formula is expressed as \( S = \frac{a(1 - r^n)}{1 - r} \), where \( S \) is the sum of the sequence, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Applying the Formula
When we want to find the sum of the first 14 terms of a sequence using the given values, we substitute \( a = -\frac{3}{2} \), \( r = -2 \), and \( n = 14 \) into the formula. This allows for an organized approach to solve for the sum.Order of Operations
To arrive at the right answer when dealing with mathematical expressions, following the order of operations is essential. Known by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), this convention ensures calculations are performed in a systematic way.
When simplifying the expression for the sum of a geometric sequence, apply PEMDAS carefully. For the given sequence, we first handle the exponent, followed by the subtraction within the parentheses and finally the division to achieve the correct result. Consistent application of the order of operations across all mathematical tasks is fundamental to accurate computation.
When simplifying the expression for the sum of a geometric sequence, apply PEMDAS carefully. For the given sequence, we first handle the exponent, followed by the subtraction within the parentheses and finally the division to achieve the correct result. Consistent application of the order of operations across all mathematical tasks is fundamental to accurate computation.
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Problem 29
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