Problem 33 - Precalculus Mathematics for Calculus

Question

Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$1,3,5,7, \dots$$

Step-by-Step Solution

Answer

The first six partial sums are 1, 4, 9, 16, 25, and 36.

Step 1: Identify the First Term

The sequence given is \(1, 3, 5, 7, \dots\). The first term \(a_1\) is \(1\).

Step 2: Determine the Pattern

The sequence follows an arithmetic pattern. Each term increases by \(2\) from the previous term. This means the sequence is written as \(a_n = 1 + (n-1)*2\) for the \(n\)-th term.

Step 3: Calculate the Partial Sum \(S_1\)

The first partial sum, \(S_1\), is just the first term of the sequence: \(S_1 = 1\).

Step 4: Calculate the Partial Sum \(S_2\)

Add the first two terms: \(S_2 = a_1 + a_2 = 1 + 3 = 4\).

Step 5: Calculate the Partial Sum \(S_3\)

Add the first three terms: \(S_3 = a_1 + a_2 + a_3 = 1 + 3 + 5 = 9\).

Step 6: Calculate the Partial Sum \(S_4\)

Add the first four terms: \(S_4 = a_1 + a_2 + a_3 + a_4 = 1 + 3 + 5 + 7 = 16\).

Step 7: Calculate the Partial Sum \(S_5\)

Add the first five terms: \(S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 1 + 3 + 5 + 7 + 9 = 25\).

Step 8: Calculate the Partial Sum \(S_6\)

Add the first six terms: \(S_6 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 1 + 3 + 5 + 7 + 9 + 11 = 36\).

Key Concepts

Understanding Arithmetic Sequences

Arithmetic sequences are simple mathematical patterns where each term increases or decreases by a constant amount from the previous one. This constant is known as the 'common difference.' In the example sequence you are working on, which is 1, 3, 5, 7, ..., the common difference is 2. This means every term is 2 units bigger than the one before it. If you need to write a general formula for this sequence, it looks like this: \(a_n = a_1 + (n-1) \times d\). Here, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) indicates the term's position in the sequence. Understanding this pattern helps you predict any term in the sequence without listing all the previous terms.

Recognizing Sequence Patterns

To correctly spot sequence patterns, you need to identify how the terms change from one to another. Let’s consider the sequence 1, 3, 5, 7, .... By subtracting consecutive terms, you see they differ by 2 each time: \(3 - 1 = 2\), \(5 - 3 = 2\), and so on. Recognizing this change, you conclude the sequence grows by 2. This is a huge step in understanding how sequences work because once you recognize a pattern, predicting future terms gets much simpler. In other words, sequence patterns are like shortcuts that lead us to the solution in a faster and easier way.

Calculating the Sum of Sequences

When you work with arithmetic sequences, you might be interested in knowing the sum of the first few terms. This is where partial sums come in. For instance, in the sequence 1, 3, 5, 7, ..., the first partial sum \(S_1\) is just the first term itself: \(1\). The second partial sum \(S_2\) is the first two terms added together: \(1 + 3 = 4\). Continuing this process gives you \(S_3 = 1 + 3 + 5 = 9\). Notice how calculating sums requires adding each term one by one until you reach the desired point. This makes partial sums a cumulative addition of terms, helping us explore the increase or total over a series of numbers.