In a number sequence, the first term is 1 and the second term is 5. From the third term onwards, each term is defined as the average (arithmetic mean) of all the preceding terms in the sequence. Under this rule, what is the value of the 25th term of the sequence?

Introduction / Context:
This is a conceptual sequence problem involving averages. The rule given is that each term from the third onward equals the arithmetic mean of all previous terms. Recognizing the pattern that emerges after the first few terms helps you avoid heavy computation, especially when asked for a far-off term like the 25th term.

Given Data / Assumptions:

• First term a₁ = 1.
• Second term a₂ = 5.
• For n ≥ 3, a_n = (a₁ + a₂ + … + a_n−1) / (n − 1).
• We must find a₂₅.

Concept / Approach:
Instead of computing all 25 terms directly, we look for a stable pattern. When each new term equals the average of all previous terms, the sequence tends to stabilize, often becoming constant from some point onward. We can compute the first few terms, look for such stabilization and then generalize.

Step-by-Step Solution:
Step 1: Calculate the first few terms explicitly. a1 = 1, a2 = 5. Step 2: For a3, use the rule: a3 = (a1 + a2) / 2 = (1 + 5) / 2 = 6 / 2 = 3. Step 3: Compute the sum S3 = a1 + a2 + a3 = 1 + 5 + 3 = 9. Step 4: For a4: a4 = S3 / 3 = 9 / 3 = 3. Step 5: New sum S4 = S3 + a4 = 9 + 3 = 12. Step 6: For a5: a5 = S4 / 4 = 12 / 4 = 3. Step 7: Now observe the pattern. Once a3 becomes 3, every new term adds 3 to the sum, and the average of all terms up to that point remains 3. Step 8: More formally, suppose for some k ≥ 3 we have a3 = a4 = … = a_k = 3 and S_k is the sum up to k. Then S_k = 1 + 5 + (k − 2)*3. Step 9: For term a_k+1, we have a_k+1 = S_k / k. Step 10: Substitute S_k: S_k = 6 + 3(k − 2) = 6 + 3k − 6 = 3k, so a_k+1 = (3k) / k = 3. Step 11: This shows by induction that all terms from a3 onward are equal to 3. Step 12: Therefore, a25 (the 25th term) must also be 3.
Verification / Alternative check:
We can compute a few more terms directly to gain confidence: a3 = 3, a4 = (1 + 5 + 3) / 3 = 3, a5 = (1 + 5 + 3 + 3) / 4 = 3. The total remains 3 times the number of terms from a3 onward, which keeps the average at 3 each time. This confirms the stabilized pattern.

Why Other Options Are Wrong:
2.5, 3.5, 4, 5: These values might appear if someone incorrectly averages only the last few terms or applies an incorrect recurrence relation. They do not match the stable pattern demonstrated by the correct inductive reasoning.
Common Pitfalls:
A major pitfall is to attempt manual computation of every term up to the 25th without spotting the stabilization after the third term, which is time-consuming and error-prone. Another is to misapply the averaging rule by taking an average of only the previous one or two terms instead of all preceding terms, which yields a different sequence altogether.

Final Answer:
The 25th term of the sequence is 3.