To determine the 77th term in the sequence $6, 14, 22, 30, \dots$, we recognize that this sequence is arithmetic because the difference between consecutive terms is constant.

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Step 1: Find the common difference $d$

The difference between consecutive terms: $d = 14 - 6 = 8$ Check the next difference: $d = 22 - 14 = 8 \quad \text{and} \quad d = 30 - 22 = 8$ Thus, the common difference $d = 8$.

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Step 2: Use the formula for the $n$-th term of an arithmetic sequence

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1) \cdot d$ Where:

• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Here:

• $a_1 = 6$,
• $d = 8$,
• $n = 77$.

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Step 3: Substitute the values

Substitute into the formula: $a_{77} = 6 + (77 - 1) \cdot 8$ Simplify step by step: $a_{77} = 6 + 76 \cdot 8$ $a_{77} = 6 + 608$ $a_{77} = 614$

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Final Answer:

The 77th term of the sequence is: $\boxed{614}$

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Do you need further clarification or a breakdown of any step? Let me know!

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Here are 5 related questions to deepen your understanding:

1. What is the 100th term of this arithmetic sequence?
2. How many terms are needed for the sequence to reach or exceed 1000?
3. What is the sum of the first 77 terms of this sequence?
4. Write the general formula for the $n$-th term of the sequence $6, 14, 22, 30, \dots$.
5. If another sequence has a first term of 10 and a common difference of 12, what is its 50th term?

Tip: In an arithmetic sequence, the difference between consecutive terms is constant, and the formula for any term relies on the first term and the common difference.