Problem 33 - Algebra and Trigonometry

Question

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ 1,3,5,7, \ldots $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of Partial Sums

Partial sums involve adding up the first "n" terms of a sequence. The nth partial sum, denoted as \( S_n \), is the sum of the first n terms of the sequence.

Step 2: Identify the Sequence Formula

The sequence given is arithmetic: \(1, 3, 5, 7, \ldots\). The common difference, \(d\), is 2. Therefore, the nth term of the sequence, \(a_n\), can be written as \(a_n = 2n - 1\).

Step 3: Calculate the First 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 Second Partial Sum \(S_2\)

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

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

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

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

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

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

Add the first five terms: \(S_5 = 1 + 3 + 5 + 7 + 9 = 25\).

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

Add the first six terms: \(S_6 = 1 + 3 + 5 + 7 + 9 + 11 = 36\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a clear pattern. Each number in the sequence is generated by adding a fixed amount to the previous number. This fixed amount is known as the "common difference". For the sequence given in the problem, the numbers are 1, 3, 5, 7, and so on.

In this sequence:

• The first term is 1.
• The second term is 3.
• The third term is 5.
• The fourth term is 7.

These terms are part of the arithmetic sequence because each term increases by the same number, which brings us to the next concept: the common difference.

Sequence Formula

To describe an arithmetic sequence mathematically, we use a formula. This formula helps us find any term in the sequence without having to list out all of the terms. The formula for the nth term (\(a_n\)) of an arithmetic sequence is:

• \( a_n = a_1 + (n-1) \, d \)

Where:

• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number you're looking for.
• \( d \) is the common difference between the terms.

For our sequence, the formula is simplified to \( a_n = 2n - 1 \) because the first term is 1 and the common difference is 2, making it easy to determine any term in the sequence.

Common Difference

The common difference in an arithmetic sequence is the amount we add (or subtract, in a decreasing sequence) from one term to get the next. It's a key part of what makes the sequence arithmetic.

• In our example sequence of 1, 3, 5, 7,..., the common difference is 2.
• We find it by subtracting the first term from the second term: \(3 - 1 = 2\).
• Consistently checking, \(5 - 3 = 2\) and \(7 - 5 = 2\), confirms this difference throughout the sequence.

Understanding the common difference helps us predict future terms and establish the structure of the sequence easily.

Nth Term

The nth term of an arithmetic sequence is the term that appears in the position "n" of the list when arranged sequentially. With an arithmetic sequence, the formula for the nth term helps us find this easily without listing all terms.

In the example sequence:

• The first term is termed as \(a_1 = 1\).
• The second term becomes \(a_2 = 3\).
• Following the formula \(a_n = 2n - 1\), to find the 5th term, plug 5 into the formula: \(a_5 = 2(5) - 1 = 9\).

By understanding the nth term, students can quickly find large numbered terms without building the entire sequence. This becomes especially useful as sequences become longer or for significantly larger n.