Recognizing Patterns - Pre Algebra

1. What is the units digit of $3^8$?

   1. The units digits of powers of 3 cycle 3, 9, 7, 1, and 8 is a multiple of 4.

2. What is the units digit of $2^{10}$?

   4. The cycle 2, 4, 8, 6 has length 4, and 10 leaves remainder 2, matching the second digit, 4.

3. A row of hexagons has perimeters 6, 10 and 14 for 1, 2 and 3 hexagons. What is the perimeter for 8 hexagons?

   34. Each added hexagon adds 4, so $6 + 4(7) = 34$.

4. The units digits of $2^1, 2^2, 2^3, \ldots$ repeat in a cycle. How many digits long is the cycle?

   4 digits: 2, 4, 8, 6, then the cycle repeats.

5. What is a sequence called when the same amount is added to get each new term?

   An arithmetic sequence. The amount added is the common difference.

6. What is the next term of the sequence $3, 6, 12, 24, \ldots$?

   48. Each term is twice the one before it.

7. A row of squares made of toothpicks uses 4, 7 and 10 toothpicks for 1, 2 and 3 squares. What expression gives the count for $n$ squares?

   $3n + 1$. Each new square adds 3 toothpicks to the first square's 4.

8. The triangular numbers begin $1, 3, 6, 10$. What is the next one?

   15. The differences are 2, 3 and 4, so the next difference is 5.

9. What is the sum of the first 5 odd numbers, $1 + 3 + 5 + 7 + 9$?

   25. The sum of the first $n$ odd numbers is $n^2$, and $5^2 = 25$.

10. Level $n$ of a dot triangle has $\frac{n(n + 1)}{2}$ dots. How many dots does level 10 have?

    55, since $\frac{10 \times 11}{2} = 55$.

11. What expression gives the $n$th term of the sequence $5, 8, 11, 14, \ldots$?

    $3n + 2$. The terms rise by 3, and $3(1) + 2 = 5$ for the first term.

12. The sequence $1, 4, 9, 16, 25, \ldots$ continues the same way. What is its 10th term?

    100. Each term is the square of its position, and $10^2 = 100$.

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