Pattern Identification in Number Series

The question requires finding the next number in the sequence: 21, 43, 85, 171, 341, ... To solve this, we need to identify the pattern governing the series.

Step-by-Step Series Solution

Step 1: Analyze Term Relationships

Let's examine the relationship between consecutive terms:

• $43 = 21 \times 2 + 1$
• $85 = 43 \times 2 - 1$
• $171 = 85 \times 2 + 1$
• $341 = 171 \times 2 - 1$

The pattern observed is to multiply the current term by 2 and then alternately add 1 and subtract 1.

The general formula can be represented as:

$Term_{n+1} = (Term_n \times 2) + (-1)^{n+1}$

Where $n$ represents the position of the term starting from $n=1$ for the operation applied to get the second term.

Step 2: Calculate the Next Term

Following the identified pattern, the next operation involves multiplying the last term (341) by 2 and adding 1 (since the last operation was subtraction):

• Next Term $= (341 \times 2) + 1$
• Next Term $= 682 + 1$
• Next Term $= 683$

Conclusion: Next Term Calculation

The next term in the series 21, 43, 85, 171, 341 is 683.