Exploring Arithmetic Sequences: A Step-by-Step Guide

Understanding arithmetic sequences is crucial for various mathematical applications and problem-solving scenarios. One of the key formulas that arithmetic sequence lovers should master is the nth term formula: [a_n a_1 (n - 1)d]

The Sequence 3, 6, 9

Letrsquo;s delve into how to use this formula to solve a specific problem. Consider the arithmetic sequence 3, 6, 9, ….

Identify the First Term ((a_1)):

The first term of our sequence, (a_1), is 3.

Determine the Common Difference ((d)):

The common difference, (d), is the difference between any two consecutive terms. For our sequence, (d 6 - 3 3).

Identify the Target Term (120):

We need to find which term in the sequence is 120. This can be represented as (a_n 120).

Solving for the Term Number ((n))

Letrsquo;s use the formula to solve for (n). [a_n a_1 (n - 1)d] Substituting the values we identified: [120 3 (n - 1) cdot 3] Simplifying the equation: begin{align*} 120 - 3 (n - 1) cdot 3 117 (n - 1) cdot 3 frac{117}{3} n - 1 39 n - 1 40 n end{align*} So, 120 is the 40th term in the sequence.

Understanding Arithmetic Progression

The sequence 3, 6, 9, 12, 15, 18, 21, … is an arithmetic progression, where each term is obtained by adding a constant (common difference) to the previous term.

Formulas and Application

The general formula for the nth term in an arithmetic sequence is:

[text{nth term} a_1 (n - 1) cdot d]

Plugging in the values for our sequence:

[text{120} 3 (n - 1) cdot 3 Rightarrow 120 - 3 (n - 1) cdot 3 Rightarrow 117 (n - 1) cdot 3 Rightarrow 39 n - 1 Rightarrow n 40]

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Conclusion

By understanding and applying the formula for finding the nth term in an arithmetic sequence, you can unravel complex mathematical problems with ease. Remember, the key to consistent success is practice and a deep understanding of the underlying concepts.