Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$8+16+32+64+\cdots$$

Step-by-Step Solution

Answer

The given sequence is a geometric sequence with a common ratio of 2. And the sum of the first 15 terms of this sequence \( S_{15} = 8 \cdot (2^{15} - 1) \).

Step 1: Identify the sequence

To identify whether the sequence is arithmetic (common difference between terms) or geometric (common ratio between terms), you check the relationship between consecutive terms. In this case, each term is twice the previous term (16/8 = 32/16 = 64/32 = 2). Therefore, this is a geometric sequence with a common ratio \( r = 2 \).

Step 2: Find the sum of the first 15 terms

Since the sequence is geometric, you use the geometric series sum formula \[ S_n = a(1 - r^n) / (1 - r) \] 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. In your case, \( a = 8, r = 2, n = 15 \). Plug in these values to the formula to calculate the sum.

Step 3: Perform the calculation

After plugging in the values, your equation becomes: \[ S_{15} = 8(1 - 2^{15}) / (1 - 2) \]. Because \(1 - 2 = -1\), the denominator becomes \(-1\), which changes the sign of the numerator. Solving the new equation, the sum of the first 15 terms is \[ S_{15} = 8 \cdot (2^{15} - 1) \] which yields the exact numerical value.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is the factor by which we multiply each term to get the next one. It is constant throughout the sequence. For the sequence given in the exercise, where the terms are 8, 16, 32, 64, and so on, the common ratio is found by dividing any term by the preceding one.
For example, dividing 16 by 8 results in 2. This same ratio applies between the pairs 32/16 and 64/32, confirming the common ratio. Thus, the common ratio for this sequence is 2. Understanding the common ratio is key to identifying and working with geometric sequences effectively, as it helps predict the growth or decline pattern of the sequence.

Geometric Sequence

A geometric sequence is a list of numbers where each number is found by multiplying the previous one by a consistent number, called the common ratio. This type of sequence can rapidly increase or decrease, depending on whether the common ratio is greater or less than 1.
In the exercise, the sequence starts at 8 and each subsequent number is obtained by multiplying the previous term by the common ratio of 2. Therefore, the sequence exhibits exponential growth. Knowing that a sequence is geometric allows us to use specific formulas to solve problems or find specific terms without listing the entire sequence. By recognizing a sequence as geometric, calculations and predictions become simpler.

Sum of Terms

When dealing with geometric sequences, finding the sum of a specific number of terms can be particularly useful, especially when the sequence is extensive. The formula for the sum of the first \( n \) terms of a geometric sequence is:

• \( S_n = a(1 - r^n) / (1 - r) \)

Here, \( a \) represents the first term of the sequence, \( r \) is the common ratio, and \( n \) is the number of terms to sum. In the problem at hand, \( a = 8 \), \( r = 2 \), and \( n = 15 \).
Substituting these values helps us solve for the sum of the first 15 terms. The calculated sum gives an overall total for those terms, illustrating the total growth from the first to the fifteenth term. This formula is vital for quickly and accurately finding sums in geometric sequences without manually adding each term.