In an arithmetic progression (A.P.), the 3rd term is 19 and the 6th term is 37. What is the value of the 13th term of this A.P.?

Introduction / Context:
This problem involves an arithmetic progression (A.P.), where each term after the first is obtained by adding a constant common difference. We are given two non consecutive terms and asked to find another later term. This tests your ability to set up equations for terms of an A.P. and solve for the first term and common difference.

Given Data / Assumptions:
Let the first term of the A.P. be a.Let the common difference be d.The 3rd term (T3) is 19.The 6th term (T6) is 37.We must find the 13th term (T13).

Concept / Approach:
In an A.P., the n-th term is given by Tn = a + (n − 1)d. Using this formula, we express T3 and T6 in terms of a and d, creating a system of two linear equations. Solving these equations yields the values of a and d, which then allow us to compute T13 directly.

Step-by-Step Solution:
Step 1: Use the formula Tn = a + (n − 1)d.Step 2: For the 3rd term, T3 = a + 2d = 19.Step 3: For the 6th term, T6 = a + 5d = 37.Step 4: Subtract the equation in Step 2 from the equation in Step 3: (a + 5d) − (a + 2d) = 37 − 19.Step 5: This simplifies to 3d = 18, so d = 6.Step 6: Substitute d = 6 into a + 2d = 19: a + 2 × 6 = 19, so a + 12 = 19 and a = 7.Step 7: Now find the 13th term: T13 = a + 12d = 7 + 12 × 6.Step 8: Compute 12 × 6 = 72, so T13 = 7 + 72 = 79.

Verification / Alternative check:
We can list some terms to check the pattern. The sequence begins as 7, 13, 19, 25, 31, 37, ... The 3rd term is 19 and the 6th term is 37, which matches the given data. Continuing with difference 6, the 7th term is 43, 8th is 49, and so on, up to the 13th term, which will indeed be 79.

Why Other Options Are Wrong:
Values like 43, 45 or 49 occur earlier in the sequence (43 as T7, 49 as T8). They are not the 13th term. Any answer smaller than 79 ignores the common difference or misapplies the term formula.

Common Pitfalls:
Some learners mistakenly plug n directly into Tn = a + nd instead of a + (n − 1)d. Others miscalculate the difference when subtracting equations, leading to an incorrect value for d. Writing the general term formula correctly and checking at least one intermediate term helps confirm the pattern.

Final Answer:
The 13th term of the arithmetic progression is 79.