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Chapter 9: Problem 41

Use mathematical induction to find a formula for the sum of the first \(n\) terms of the sequence. $$1,5,9,13, \ldots$$

Short Answer

Expert verified
The formula for the sum of the first n terms of the sequence 1,5,9,13,... is \(\frac{n}{2}(8 + 4(n - 1))\). This formula has been proven by mathematical induction.

Step by step solution

01

Determine the formula for the nth term

The formula for the nth term of an arithmetic sequence is given by \(a + (n - 1) * d\), where a is the first term, d is the common difference, and n is the term number. For our sequence, a = 1 and d = 4. So the nth term is \(1 + 4(n - 1)\), simplifying to \(4n - 3\).
02

Assume the formula is true for n=k

Assume that the formula for the sum of the first k terms (denoted as \(S_k\)) is \(\frac{k}{2}(8 + 4(k - 1))\). This is the step of induction where we assume the statement is true for some arbitrary positive integer k.
03

Prove the formula is true for n=k+1

Now we need to prove that the formula holds for \(k+1\). So if we add the \((k+1)\)th term to \(S_k\), it should equal to \(\frac{(k+1)}{2}(8 + 4((k + 1) - 1))\). The \((k+1)\)th term is \(4(k + 1) - 3 = 4k + 4 - 3 = 4k + 1\). After adding this to \(S_k\), we get \(\frac{k}{2}(8 + 4(k - 1)) + 4k + 1\). Simplify it to get \(\frac{(k+1)}{2}(8 + 4((k + 1) - 1))\). This shows that if the formula holds for \(k\), then it also holds for \(k+1\).
04

Verification for n=1

Check the base case when n = 1 to start off the process of mathematical induction. The actual first term of the sequence when n=1 is 1. As per the formula, \(S_1\) would become \(\frac{1}{2}(8 + 4(1 - 1)) = 1\). Hence, the formula holds for n = 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms remains constant. This constant difference is key to identifying and working with arithmetic sequences. In the sequence you're analyzing, namely 1, 5, 9, 13, ..., this difference is 4. You'll notice that for each step from one term to the next, you simply add 4.
  • First term (\(a = 1\))
  • Common difference (\(d = 4\))
This type of sequence is straightforward, and once you know the first term and the common difference, you can easily predict any term in the sequence using the nth term formula.
Inductive Hypothesis
The inductive hypothesis is a crucial part of mathematical induction, where you assume a statement is true for some integer k. For instance, in finding the sum of the first n terms of the given arithmetic sequence, you assume that a formula works correctly for the first k terms. The formula for this assumed step is important to preserve the consistency of your proof.
  • The assumption helps us to work out the logic between two steps.
  • This involves assuming the formula works for \(S_k\), the sum of the first k terms.
By assuming the hypothesis holds, it sets the stage to prove it for the next stage, \(n = k + 1\). This practice is key to linking each step in mathematical induction.
Base Case
The base case is the initial step in a mathematical induction proof, which verifies that the statement is true for the first value in the sequence, generally \(n = 1\). Establishing the base case is like setting the foundation for a building; it ensures everything else rests on a solid premise.
For this particular arithmetic sequence, the base case confirms that the sum of the first term is equal to the actual first term of the sequence. Essentially, you check that your formula \(S_1\) gives you the expected outcome, which establishes confidence to extend the argument further. If the base case fails, the entire inductive step fails.
nth Term Formula
To find any term in an arithmetic sequence, you use the nth term formula. This formula is derived from understanding the sequence's consistent step or difference between terms. The general form for an arithmetic sequence is \(a + (n - 1) imes d\), where a is the first term, and d is the common difference.
  • Apply this formula to find the nth term of your sequence.
  • For example, to find the nth term of the sequence like 1, 5, 9, 13, the formula becomes \(1 + (n - 1) imes 4\).
Simplifying the formula gives you \(4n - 3\), allowing you to plug in any number for n to get its corresponding term in the sequence. Mastering this formula helps in deeper analysis like deriving the sum of sequence terms.

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