Math worksheet featuring number sequences and pattern recognition exercises.

Here are the solutions to the problems on the worksheet.

1. Write down the next two terms.

a. 2, 7, 12, 17, \_\_\_, \_\_\_
* Logic: Add 5 to each number ($2+5=7$, $7+5=12$).
* Next term: $17 + 5 = 22$
* Term after that: $22 + 5 = 27$

b. 100, 96, 92, 88, \_\_\_, \_\_\_
* Logic: Subtract 4 from each number ($100-4=96$).
* Next term: $88 - 4 = 84$
* Term after that: $84 - 4 = 80$

c. 13, 25, 37, 49, \_\_\_, \_\_\_
* Logic: Add 12 to each number ($13+12=25$).
* Next term: $49 + 12 = 61$
* Term after that: $61 + 12 = 73$

d. 2, 6, 18, 54, \_\_\_, \_\_\_
* Logic: Multiply by 3 ($2\times3=6$, $6\times3=18$).
* Next term: $54 \times 3 = 162$
* Term after that: $162 \times 3 = 486$

e. 2, 3, 5, 8, 12, \_\_\_, \_\_\_
* Logic: The amount you add increases by 1 each time.
* $2 + \mathbf{1} = 3$
* $3 + \mathbf{2} = 5$
* $5 + \mathbf{3} = 8$
* $8 + \mathbf{4} = 12$
* Next step is adding 5: $12 + 5 = 17$
* Step after that is adding 6: $17 + 6 = 23$

f. -2, 4, 10, 16, \_\_\_, \_\_\_
* Logic: Add 6 to each number ($-2+6=4$).
* Next term: $16 + 6 = 22$
* Term after that: $22 + 6 = 28$

g. 1, 2, 4, 8, 16, \_\_\_, \_\_\_
* Logic: Multiply by 2 (doubling) each time.
* Next term: $16 \times 2 = 32$
* Term after that: $32 \times 2 = 64$

h. 1, 4, 9, 16, \_\_\_, \_\_\_
* Logic: These are square numbers ($1^2, 2^2, 3^2, 4^2$). Alternatively, add odd numbers (+3, +5, +7...).
* Next term: $5^2 = 25$
* Term after that: $6^2 = 36$

i. 3, 9, 27, 81, \_\_\_, \_\_\_
* Logic: Multiply by 3 ($3\times3=9$, $9\times3=27$).
* Next term: $81 \times 3 = 243$
* Term after that: $243 \times 3 = 729$

j. 1, 5, 25, 125, \_\_\_, \_\_\_
* Logic: Multiply by 5 ($1\times5=5$, $5\times5=25$).
* Next term: $125 \times 5 = 625$
* Term after that: $625 \times 5 = 3125$

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2. Find a formula for the $n^{th}$ term.

To find the formula for arithmetic sequences (where you add or subtract the same number), use the pattern: $(\text{difference} \times n) + \text{adjustment}$.

a. 3, 7, 11, ...
* Difference is +4. So start with $4n$.
* For $n=1$, $4(1)=4$. We need 3, so subtract 1.
* Formula: $4n-1$

b. 5, 11, 17, ...
* Difference is +6. Start with $6n$.
* For $n=1$, $6(1)=6$. We need 5, so subtract 1.
* Formula: $6n-1$

c. 13, 20, 27, ...
* Difference is +7. Start with $7n$.
* For $n=1$, $7(1)=7$. We need 13, so add 6.
* Formula: $7n+6$

d. 80, 76, 72, ...
* Difference is -4. Start with $-4n$.
* For $n=1$, $-4(1)=-4$. We need 80, so add 84.
* Formula: $-4n+84$

e. 7, 10, 13, 16, ...
* Difference is +3. Start with $3n$.
* For $n=1$, $3(1)=3$. We need 7, so add 4.
* Formula: $3n+4$

f. 10, 6, 2, -2, ...
* Difference is -4. Start with $-4n$.
* For $n=1$, $-4(1)=-4$. We need 10, so add 14.
* Formula: $-4n+14$

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Final Answer:

1. Next two terms:
a. 22, 27
b. 84, 80
c. 61, 73
d. 162, 486
e. 17, 23
f. 22, 28
g. 32, 64
h. 25, 36
i. 243, 729
j. 625, 3125

2. Formula for the $n^{th}$ term:
a. $4n-1$
b. $6n-1$
c. $7n+6$
d. $-4n+84$
e. $3n+4$
f. $-4n+14$