Question

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \ldots\)

Short answer

The next two terms of the sequence are 1/16 and -1/32.

Step-by-step solution

Identify the pattern

Look at the terms in the sequence. They alternate between positive and negative, and each term is half the absolute value of the previous term. So, the pattern of the sequence is: multiply the previous term by -1/2.

Apply the pattern to find the next term

The last term given in the sequence is -1/8. Multiply this term by -1/2. The result is \(\frac{1}{16}\) which is the next term.

Repeat the pattern to find the next next term

Again, multiply the last found term (1/16) by -1/2. The result is \(-\frac{1}{32}\) which is the next term in the sequence.

Key concepts

Understanding Convergent Series

Convergent series are a pivotal concept within mathematical sequences, particularly when diving into infinite series. Essentially, a series is convergent if the sum of its terms approaches a specific value as the number of terms increases. In other words, if you keep adding term after term, the total will settle towards a finite number, rather than escalating to infinity or oscillating indefinitely.

An intuitive way to envision a convergent series is to think of adding smaller and smaller pieces to a sum; imagine a pile of sand where each new grain is a fraction of the size of the preceding one. Eventually, the pile will grow very slowly and reach a limit—a point at which adding more grains of sand hardly increases the size of the pile.

Arithmetic Sequence Simplified

Moving on to an arithmetic sequence, this is one of the simplest types of sequences to understand. An arithmetic sequence features a constant difference between consecutive terms. This difference is known as the common difference, and can be any real number. The arithmetic sequence is linear, just like a straight line on a graph.

To find subsequent terms in an arithmetic sequence, simply add (or subtract, if the common difference is negative) the common difference from the last term. For instance, in an arithmetic sequence starting with 1 where each term increases by 3, the first few terms would be 1, 4, 7, 10, and so forth. It's a very straightforward and predictable pattern.

Decoding Geometric Sequences

On the flip side, we have geometric sequences which operate on a multiplicative pattern, rather than an additive one. In these sequences, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric sequences can visually resemble exponential growth or decay, depending on the value of the common ratio.

For example, if you start with the number 2 and your common ratio is 3, the sequence begins 2, 6, 18, 54, and so on. The terms get progressively larger (or smaller if the common ratio is between 0 and 1) at a rapid rate—which is a stark contrast to the steady increase or decrease seen in arithmetic sequences.