Let's analyze the sequences in the image and determine the next two terms for each.

Sequence (i): 1, 4, 7, 10, 13...

This sequence has a constant difference of 3 between consecutive terms, making it an arithmetic progression.

• Formula: $a_n = a_1 + (n-1)d$, where $a_1 = 1$ and $d = 3$.

The next two terms after $13$ are: $13 + 3 = 16$ $16 + 3 = 19$

So, the next terms are 16, 19.

Sequence (ii): 2, 6, 18, 54...

Each term is obtained by multiplying the previous term by $3$, making it a geometric progression.

• Formula: $a_n = a_1 \cdot r^{n-1}$, where $a_1 = 2$ and $r = 3$.

The next two terms after $54$ are: $54 \cdot 3 = 162$ $162 \cdot 3 = 486$

So, the next terms are 162, 486.

Sequence (iii): 1, 3, 4, 7, 18, 29, 47...

This sequence does not follow a straightforward arithmetic or geometric pattern. Let's analyze differences between terms:

• Differences: $3 - 1 = 2$, $4 - 3 = 1$, $7 - 4 = 3$, $18 - 7 = 11$, $29 - 18 = 11$, $47 - 29 = 18$.

Observations: The differences appear to alternate between a smaller number (2, 1, 3) and a larger one (11, 18). We need more context for the pattern.

Can you clarify or confirm if the differences are alternating in pairs? If yes, the next differences would follow $11$ and $18$, making the next terms: $47 + 11 = 58$ $58 + 18 = 76$

Next terms may be 58, 76. Let me know if this matches what you expect!

Would you like further clarification or details on any part? 😊

Related Questions:

1. How can arithmetic and geometric progressions be represented graphically?
2. What is the difference between arithmetic, geometric, and recursive sequences?
3. Could the third sequence be generated by a recursive formula?
4. How do you derive general formulas for unknown sequences?
5. Are there real-world applications of such sequences in physics or economics?

Tip:

For mixed patterns (like in sequence iii), analyzing differences or ratios between terms often reveals hidden structures.