In the puzzle sequence 1, 11, 21, 1211, 111221, what is the next term that should come after 111221?

Introduction / Context:
This question tests your ability to recognise and extend a famous pattern called the "look-and-say" sequence. Unlike simple arithmetic progressions or geometric progressions, here each term is generated by verbally describing the digits of the previous term. Understanding how to read off the previous number and convert that description back into digits is the key skill needed to find the next term after 111221.

Given Data / Assumptions:

• The given sequence of terms is: 1, 11, 21, 1211, 111221.
• We are asked to determine the next term that follows 111221.
• Each term is formed from the previous term by "looking" at groups of identical digits and then "saying" how many of each digit there are.
• The answer must be selected from the provided options.
• We assume the pattern has no hidden arithmetic; it is purely descriptive.

Concept / Approach:
In the look-and-say sequence, you create the next term by reading the previous term digit by digit, counting consecutive identical digits, and then writing down the count followed by the digit. For example, the term 21 is read as "one 2, one 1", which becomes 1211. The same rule is applied repeatedly. Our task is to apply this description rule to the term 111221 in order to generate the next term.

Step-by-Step Solution:
Step 1: Write down the last given term: 111221.Step 2: Group consecutive identical digits: 111, 22, 1.Step 3: Read each group out loud: "three 1s", "two 2s", "one 1".Step 4: Convert this spoken description back into digits by writing the count followed by the digit: "three 1s" becomes 31, "two 2s" becomes 22, and "one 1" becomes 11.Step 5: Concatenate these parts: 31 + 22 + 11 = 312211. Thus, the next term is 312211.

Verification / Alternative check:
You can quickly verify the rule by applying it to earlier terms. Starting from 1, "one 1" becomes 11. From 11, "two 1s" becomes 21. From 21, "one 2, one 1" becomes 1211. From 1211, "one 1, one 2, two 1s" becomes 111221. This confirms that the same consistent rule has been used, so applying it to 111221 to get 312211 is correct.

Why Other Options Are Wrong:
Option 132112 does not correctly encode the groups in 111221, since it suggests different counts and ordering. Option 111222 treats the sequence as simply increasing digits rather than describing the previous term. Option 213211 also mixes counts and digits incorrectly. Only option 312211 faithfully represents "three 1s, two 2s, one 1".

Common Pitfalls:
Learners sometimes misread the groups, for example interpreting 111221 as "two 1s, one 1, two 2s, one 1" rather than grouping identical digits consecutively as 111, 22, 1. Another common error is to try to find an arithmetic pattern in the values themselves rather than seeing that the sequence is descriptive. Carefully grouping digits and strictly following the count-then-digit rule avoids these mistakes.

Final Answer:
The correct next term in the sequence is 312211.