Identify Number Sequences - International Brotherhood of Electrical Workers Exam IBEW

1. Which option correctly identifies an arithmetic sequence: $1, 2, 4, 8$ or $1, 4, 7, 10$?

   $1, 4, 7, 10$ is arithmetic. The sequence $1, 4, 7, 10$ has constant differences of 3, unlike the exponential growth in $1, 2, 4, 8$.

2. What is the next term in the arithmetic sequence $\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \dots$?

   $-\frac{5}{2}$. Decreasing by 1 (or -1) each step, subtracting 1 from -$\frac{3}{2}$ yields the next fraction.

3. What is the next term in the arithmetic sequence $0.5, 1.0, 1.5, 2.0, \dots$?

   $2.5$. The sequence increments by 0.5 each time, so adding 0.5 to 2.0 gives the subsequent term.

4. What is the defining property of an arithmetic sequence in terms of consecutive terms?

   Consecutive terms differ by a constant $d$. This property ensures each term is obtained by adding a fixed value to the previous one, distinguishing arithmetic sequences.

5. What is the recursive definition of an arithmetic sequence using $a_1$ and $d$?

   $a_1$ given; $a_n = a_{n-1} + d$. The recursive form starts with the initial term and generates each subsequent term by adding the common difference.

6. Identify the common difference $d$ for the sequence $-4, -1, 2, 5, \dots$.

   $d = 3$. Starting from -4, adding 3 repeatedly yields the subsequent terms, confirming the positive common difference.

7. What is the first term $a_1$ if $a_5 = 21$ and the common difference is $d = 4$?

   $a_1 = 5$. Solving $a_5 = a_1 + 4d$ for $a_1$ by subtracting $4 \times 4$ from 21 determines the initial term.

8. Identify the common difference $d$ for the sequence $12, 9, 6, 3, \dots$.

   $d = -3$. Each term decreases by 3, establishing a constant negative difference in this decreasing arithmetic sequence.

9. Identify whether $a_n = 2n^2 + 1$ defines an arithmetic sequence.

   No; differences are not constant. The quadratic term $n^2$ results in increasing differences (e.g., 6, 10), preventing constant differences.

10. What is $a_8$ for the arithmetic sequence $a_n = 10 + (n-1)(-2)$?

    $a_8 = -4$. Substituting $n=8$ into the given explicit formula yields the eighth term directly.

11. Identify the missing term: $\frac{1}{3}, \square, \frac{7}{3}$ is arithmetic. What is $\square$?

    $\frac{4}{3}$. As the arithmetic mean of $\frac{1}{3}$ and $\frac{7}{3}$, it ensures equal differences in the three-term sequence.

12. Identify the missing term: $4, 9, \square, 19$ is arithmetic. What is $\square$?

    $14$. To maintain a constant difference, the average of 9 and 19 fills the gap, or incrementally add 5 from 9.

13. What is the constant second term minus first term test for arithmetic using $a_1, a_2, a_3$?

    Arithmetic if $a_2 - a_1 = a_3 - a_2$. For the first three terms, equal differences indicate the sequence maintains a constant rate throughout.

14. Identify whether $a_n = 7 - 5n$ defines an arithmetic sequence, and state $d$.

    Yes; $d = -5$. The linear form $a_n = 7 - 5n$ produces constant differences of -5, qualifying it as arithmetic.

15. Identify the missing term: $-10, \square, -2, 2$ is arithmetic. What is $\square$?

    $-6$. The common difference of 4 places -6 between -10 and -2 to ensure equal steps to 2.

16. Identify whether $2, 5, 9, 14, \dots$ is arithmetic.

    No; differences are not constant. The differences increase progressively (3, 4, 5), violating the constant difference requirement for arithmetic sequences.

17. Which option is the common difference for $-1, 2, 5, 8$?

    $d = 3$. Consistent differences of 3 between consecutive terms confirm the common difference in this increasing sequence.

18. What is the common difference $d$ in an arithmetic sequence, using terms $a_n$ and $a_{n-1}$?

    $d = a_n - a_{n-1}$. The common difference is calculated by subtracting the previous term from the current term in an arithmetic sequence.

19. Identify whether $8, 8, 8, 8, \dots$ is arithmetic, and state the common difference.

    Yes; $d = 0$. Constant terms imply a common difference of zero, fitting the definition of an arithmetic sequence.

20. Identify the $n$th term: sequence $6, 2, -2, -6, \dots$ written as $a_n$.

    $a_n = 6 + (n-1)(-4)$. With first term 6 and common difference -4, the explicit formula expresses any term in the sequence.

21. What is $a_6$ for the arithmetic sequence with $a_1 = 2$ and $d = 3$?

    $a_6 = 17$. Applying the explicit formula $a_n = 2 + (n-1) \times 3$ for $n=6$ computes the sixth term.

22. What is the next term in the arithmetic sequence $5, 1, -3, -7, \dots$?

    $-11$. With a common difference of -4, subtracting 4 from -7 produces the next term in the sequence.

23. Identify whether $3, 7, 11, 15, \dots$ is arithmetic.

    Yes; $d = 4$. The sequence increases by 4 each time, maintaining a constant difference characteristic of arithmetic sequences.

24. What is the common difference $d$ if $a_1 = -3$ and $a_6 = 12$ in an arithmetic sequence?

    $d = 3$. Using $a_6 = a_1 + 5d$, solving for $d$ gives the constant added five times to reach 12 from -3.

25. What is the explicit formula for the $n$th term of an arithmetic sequence with first term $a_1$ and difference $d$?

    $a_n = a_1 + (n-1)d$. The formula derives the $n$th term by adding the common difference $(n-1)$ times to the first term.

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