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Chapter 10: Problem 37

Find the indicated term using the information given. $$a_{1}=5, d=4 ; \text { find } a_{15}$$

Short Answer

Expert verified
The 15th term, \( a_{15} \), is 61.

Step by step solution

01

Understand the Arithmetic Sequence Formula

To find the 15th term of an arithmetic sequence, we use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n - 1) imes d \). Here, \( a_1 \) is the first term and \( d \) is the common difference.
02

Identify the Given Values

Identify and substitute the given values into the formula: \( a_1 = 5 \), \( d = 4 \), and \( n = 15 \). We substitute these values into the formula to find \( a_{15} \).
03

Substitute Values into Formula

Substitute the values into the formula: \( a_{15} = 5 + (15 - 1) imes 4 \).
04

Simplify the Expression

Calculate the expression inside the parentheses: \( 15 - 1 = 14 \). So the formula becomes \( a_{15} = 5 + 14 imes 4 \).
05

Multiply and Add

Multiply 14 by 4 to get 56, then add this to 5. This gives you \( a_{15} = 5 + 56 = 61 \).

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Key Concepts

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

Understanding the nth term formula in arithmetic sequences
The nth term formula in arithmetic sequences is a powerful tool. With it, you can find any term in the sequence without writing out all the preceding terms. The formula is given by: \( a_n = a_1 + (n - 1) \times d \). Here, \( a_n \) represents the nth term that you want to find.
  • \( a_1 \): This is the first term of the sequence.
  • \( n \): This is the position of the term in the sequence.
  • \( d \): This stands for the common difference.
The formula works by starting at the first term and taking steps of the common difference until it reaches the nth position. Each step is equivalent to adding the common difference, \( d \), to the previous term. This formula simplifies the process significantly, especially when n is large. So, always keep this formula in mind when dealing with arithmetic sequences.
Deciphering the common difference
The common difference in an arithmetic sequence is crucial because it dictates the rate at which the sequence progresses. In simple words, it is the number added to each term to arrive at the next term. Suppose your sequence starts from a first term, \( a_1 \). To find the next term, you simply add the common difference.
  • For example, if \( a_1 = 5 \) and \( d = 4 \), then the second term will be \( 5 + 4 = 9 \).
  • The third term will be \( 9 + 4 = 13 \).
Hence, the sequence develops based on constant differences between consecutive terms. The common difference can be positive, negative, or even zero. If positive, the sequence is increasing. If negative, it is decreasing. If zero, all terms are the same.
The significance of the first term
In any arithmetic sequence, the first term, \( a_1 \), sets the stage for all subsequent terms. It refers to the starting point or the initial value of the sequence. Knowing \( a_1 \) is necessary because the formula for finding any term in the sequence relies on it.
  • Imagine \( a_1 \) as the anchor of a ship; it determines the starting position before the sequence sails through additional terms.
  • Without it, the sequence could float anywhere without a clear path.
When represented in the nth term formula, \( a_1 \) acts as the base from which additions are made using the common difference, \( d \). It is from this foundational term that all other terms are derived, making it indispensable in understanding the entire sequence.

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