5 6 27 160 1169 ?

Steps to Solve

1. Identify the first difference between consecutive terms

Calculate the difference between each pair of consecutive numbers:

• $6 - 5 = 1$
• $27 - 6 = 21$
• $160 - 27 = 133$
• $1169 - 160 = 1009$

The differences are: $1, 21, 133, 1009$.

2. Identify differences of differences

Now, calculate the differences of the differences:

• $21 - 1 = 20$
• $133 - 21 = 112$
• $1009 - 133 = 876$

The second level differences are: $20, 112, 876$.

3. Identify third level differences

Calculate the differences of the second level differences:

• $112 - 20 = 92$
• $876 - 112 = 764$

The third level differences are: $92, 764$.

4. Pattern identification of differences

Observe that each difference seems to be progressively increasing. Noticing a pattern, we can find a relationship:

• The first terms ($5, 6, 27, 160, 1169$) can also be generated by a mathematical relationship involving multiplication and addition.

5. Creating a formula

If we observe closely:

• $6 = 5 \times 1 + 1$
• $27 = 6 \times 4 + 3$
• $160 = 27 \times 5 + 5$
• $1169 = 160 \times 7 + 9$

Following this pattern in the next term, we can deduce the multiplier and addition:

• It appears we can multiply by 8 and then add 13 to the last number.

Let's calculate:

$$ 1169 \times 8 + 13 = 9352 + 13 = 9365 $$

6. Final answer extraction

Thus, the next number in the series should be $9365$.