In an arithmetic progression, the sum of the 3rd and 15th terms is equal to the sum of the 6th, 11th and 13th terms. Under this condition, which term of the series must necessarily be equal to zero?

Introduction / Context:
This question is about arithmetic progressions (A.P.) and tests your ability to express terms algebraically and use given conditions to deduce properties of the progression. Specifically, you must determine which term becomes zero when a certain relationship between sums of terms is given.

Given Data / Assumptions:
- We have an arithmetic progression with first term a and common difference d.
- The sum of the 3rd and 15th terms equals the sum of the 6th, 11th and 13th terms.
- We must find which term of this A.P. is necessarily equal to 0.

Concept / Approach:
In an A.P., the nth term is given by:
a_n = a + (n - 1) * d.
We will write expressions for the 3rd, 15th, 6th, 11th and 13th terms in terms of a and d. Then we impose the given equality, simplify to find a relationship between a and d, and finally determine for which n the term a_n equals zero.

Step-by-Step Solution:
Step 1: Write the required terms:a_3 = a + 2d, a_15 = a + 14d.Step 2: Sum of 3rd and 15th terms: a_3 + a_15 = (a + 2d) + (a + 14d) = 2a + 16d.Step 3: Write the other three terms: a_6 = a + 5d, a_11 = a + 10d, a_13 = a + 12d.Step 4: Sum of 6th, 11th and 13th terms: a_6 + a_11 + a_13 = (a + 5d) + (a + 10d) + (a + 12d) = 3a + 27d.Step 5: Given condition: 2a + 16d = 3a + 27d.Step 6: Rearrange: 2a + 16d - 3a - 27d = 0 → -a - 11d = 0 → a = -11d.Step 7: General term a_n = a + (n - 1)d = -11d + (n - 1)d = (n - 12)d.Step 8: For a_n to be zero and d ≠ 0, we need (n - 12) = 0 → n = 12.

Verification / Alternative check:
By taking a specific value for d, say d = 1, we get a = -11. Then the 12th term is a_12 = -11 + 11 = 0, and you can numerically verify that the sum of the 3rd and 15th terms equals the sum of the 6th, 11th and 13th terms, confirming the algebraic derivation.

Why Other Options Are Wrong:
Terms like the 7th, 9th, 14th or 5th would give a_n = (n - 12)d ≠ 0 for n ≠ 12 (as long as d ≠ 0). They do not satisfy the requirement of being forced to zero by the given condition. Only the 12th term is guaranteed to be zero for any non-zero common difference d consistent with the condition.

Common Pitfalls:
Some learners forget the general term formula or miscalculate one of the indices, for example taking a_15 = a + 15d instead of a + 14d. Others incorrectly add the terms or simplify the equation, leading to a wrong relationship between a and d. Carefully writing each term and performing algebraic steps slowly avoids these mistakes.

Final Answer:
The 12th term of the arithmetic progression must be equal to zero.