To find the next term in the series $2, 14, 32, 56, \,?$, let's identify the pattern.

Step 1: Identify the Differences Between Terms

Let's calculate the difference between each consecutive term:

1. $14 - 2 = 12$
2. $32 - 14 = 18$
3. $56 - 32 = 24$

The differences between the terms are $12, 18,$ and $24$.

Step 2: Determine the Pattern in the Differences

Now, let's find the pattern in the differences:

• The first difference is $12$.
• The second difference is $18$. (This is $12 + 6$)
• The third difference is $24$. (This is $18 + 6$)

The pattern is that the differences increase by $6$ each time.

Step 3: Find the Next Difference

The next difference should be:

$24 + 6 = 30$

Step 4: Find the Next Term in the Series

Now, add this difference to the last term:

$56 + 30 = 86$

Answer

The next term in the series is 86.

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Would you like further details or have any questions? Here are some related queries:

1. How to identify patterns in arithmetic sequences?
2. What are common types of sequences in mathematics?
3. How to find the nth term of an arithmetic sequence?
4. What are the differences between arithmetic and geometric sequences?
5. How to solve more complex series with variable patterns?

Tip: Always look for patterns in the differences or ratios between terms to identify the sequence type!