In an arithmetic progression, the 3rd term is 13 and the 5th term is 21. What is the 13th term of this progression?

Introduction / Context:
This problem deals with arithmetic progressions (A.P.), which are sequences where each term after the first is obtained by adding a constant difference. You are given the 3rd and 5th terms of an A.P. and asked to find the 13th term. This tests your ability to use the general term formula of an arithmetic progression and to solve for the first term and common difference from partial information.

Given Data / Assumptions:

- Let the first term of the A.P. be a. - Let the common difference be d. - The 3rd term is given as 13. - The 5th term is given as 21. - We need to find the 13th term.

Concept / Approach:
In an arithmetic progression, the n-th term T_n is given by T_n = a + (n - 1)d. Using this formula, we can express the 3rd and 5th terms in terms of a and d, giving two equations. Solving these equations yields the values of a and d. Then we substitute these into the formula for the 13th term to find its value. This is a standard method in A.P. questions.

Step-by-Step Solution:
Step 1: Write the formula for the n-th term: T_n = a + (n - 1)d. Step 2: For the 3rd term, n = 3, so T_3 = a + 2d. We are given T_3 = 13, so a + 2d = 13. Step 3: For the 5th term, n = 5, so T_5 = a + 4d. We are given T_5 = 21, so a + 4d = 21. Step 4: Subtract the first equation from the second to eliminate a: (a + 4d) - (a + 2d) = 21 - 13. Step 5: This gives 2d = 8, so d = 4. Step 6: Substitute d = 4 back into a + 2d = 13: a + 2 * 4 = 13. Step 7: So a + 8 = 13, which gives a = 5. Step 8: Now find the 13th term using T_13 = a + 12d. Step 9: Substitute a = 5 and d = 4: T_13 = 5 + 12 * 4. Step 10: Compute 12 * 4 = 48, so T_13 = 5 + 48 = 53.

Verification / Alternative check:
We can list some terms of the progression to verify. With a = 5 and d = 4, the sequence starts as 5, 9, 13, 17, 21, ... The 3rd term is 13 and the 5th term is 21, matching the given information. Continuing, the 6th term is 25, and so on. The 13th term is obtained by adding 4 twelve times to 5, which is 5 + 48 = 53, consistent with our formula based calculation.

Why Other Options Are Wrong:
Values like 49, 57 or 61 would arise from incorrect values of the common difference or arithmetic mistakes when computing T_13. For example, using d = 3 or miscounting the number of increments would lead to such errors. Only 53 is consistent with both given terms and the arithmetic progression formula.

Common Pitfalls:
Common mistakes include miswriting the general term formula as a + nd instead of a + (n - 1)d, or incorrectly subtracting equations and getting d wrong. Another pitfall is miscounting the steps to the 13th term and using 11d instead of 12d. Carefully applying the formula and double checking each algebraic step prevents these issues.

Final Answer:
The 13th term of the arithmetic progression is 53, which corresponds to option A.