Problem 27 - Algebra and Trigonometry

Question

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 11 terms of the geometric sequence: \(3,-6,12,-24, \dots\)

Step-by-Step Solution

Answer

The sum of the first 11 terms of the geometric sequence is -2047.

Step 1: Identify a, r, and n

a is the first term of the sequence, so a = 3. r is the ratio between any term and the preceding term. For this sequence, divide any term to the preceding term, so r = -6/3 = -2. n is the number of terms, which the question specified to be 11. So a = 3, r = -2, and n = 11.

Step 2: Substitute a, r, and n into the formula

Substitute a = 3, r = -2, and n = 11 into the sum of a geometric sequence formula S = a(1 - r^n) / (1 - r). This gives S = 3(1 - (-2)^11) / (1 - (-2)).

Step 3: Solve the equation

\(-2^11= -2^{(b1 + 10)}, where b1=1 and 10 is an even number so the following is even, and negative to the power of even number is positive, hence it turns into \(2^{(1+2*5)}=2^{11}=2048\). This gives S = 3(1 - 2048) / (1 + 2) = 3 * -2047 / 3 = -2047

Key Concepts

Geometric Sequence Formula

Understanding the geometric sequence formula is crucial for students when tackling problems related to geometric sequences. A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

The formula for the sum of the first n terms of a geometric sequence is given by:
\( S_n = \frac{a(1 - r^n)}{1 - r} \), where \( S_n \) is the sum of the first n terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms to be added.

In the given exercise, we use this formula to find the sum of the first 11 terms. By identifying the first term (\( a = 3 \) ) and the common ratio (\( r = -2 \)) from the sequence, we can plug these values into the formula to calculate the sum. The sign of the ratio (\( -2 \) ) results in the alternating positive and negative pattern of the sequence.

Arithmetic Operations in Sequences

When performing arithmetic operations in sequences, it's essential to recognize patterns and apply the right formulas. In a geometric sequence, identifying the fixed pattern the ratio (\( r \) ) creates as the sequence progresses allows for the simplification of expressions during calculation.

In step 3 of the solution, the exponent operation plays a significant role: \( r^n = (-2)^n \). Since n is 11 (an odd number), \( (-2)^n \) maintains a negative sign. Had n been even, the result would have turned positive. Arithmetic operations like these can fundamentally alter the results in sequences, making careful calculation paramount.

Moreover, division by \( (1 - r) \) in the formula is an arithmetic step that should never be overlooked, as it provides the denominator necessary to complete the sum calculation of the geometric sequence.

Converging Series

Converging series are a fascinating concept within the realm of sequences and series. A series is said to be converging if the sum of its terms approaches a finite value as n approaches infinity. In the context of geometric sequences, a converging series occurs when the absolute value of the common ratio \( |r| < 1 \).

The series in our exercise is a diverging series because \( |r| = |-2| = 2 > 1 \), meaning the sum does not approach a finite limit. Conversely, if the ratio had been between -1 and 1, the terms would have gotten closer and closer to zero, leading to a finite sum. Understanding the convergence of series is key to determining the behavior of infinite sequences and the sum of their terms.