Pattern Analysis

Examine the differences between consecutive terms in the sequence: 3, 6, 11, 20, 33, 70.

• $6 - 3 = 3$
• $11 - 6 = 5$
• $20 - 11 = 9$
• $33 - 20 = 13$
• $70 - 33 = 37$

The sequence of differences is: 3, 5, 9, 13, 37.

Second Difference Identification

Calculate the differences between consecutive terms in the difference sequence (second differences):

• $5 - 3 = 2$
• $9 - 5 = 4$
• $13 - 9 = 4$
• $37 - 13 = 24$

The second differences are: 2, 4, 4, 24.

A common pattern involves powers of 2. Let's test if the intended second differences follow the pattern 2, 4, 8, 16...

Sequence Reconstruction

Assuming the second differences are powers of 2 (2, 4, 8, 16,...), the sequence of differences should be:

• 1st difference: 3
• 2nd difference: $3 + 2^1 = 5$
• 3rd difference: $5 + 2^2 = 9$
• 4th difference: $9 + 2^3 = 17$
• 5th difference: $17 + 2^4 = 33$

This suggests the correct difference sequence is 3, 5, 9, 17, 33.

Now, reconstruct the original sequence using these differences:

• Term 1: 3
• Term 2: $3 + 3 = 6$
• Term 3: $6 + 5 = 11$
• Term 4: $11 + 9 = 20$
• Term 5: $20 + 17 = 37$
• Term 6: $37 + 33 = 70$

The pattern generates the sequence: 3, 6, 11, 20, 37, 70.

Conclusion

Comparing the generated sequence (3, 6, 11, 20, 37, 70) with the given sequence (3, 6, 11, 20, 33, 70), the number 33 does not fit the pattern.