If 8, p, q and 26 is an A.P. Find the values of p and q

Analysis

The question asks for the missing terms ($p$ and $q$) in an Arithmetic Progression (A.P.) sequence: $8, p, q, 26$. In an A.P., the difference between consecutive terms is constant. This constant is called the common difference ($d$).

Concept

An Arithmetic Progression follows the formula: $T_n = a + (n - 1)d$ Where:

• $a$ is the first term.
• $d$ is the common difference.
• $n$ is the position of the term.

Given the sequence $8, p, q, 26$:

• $T_1 = a = 8$
• $T_4 = 26$

Step-by-Step Solution

Step 1: Find the common difference ($d$) Using the formula for the 4th term ($T_4$): $T_4 = a + (4 - 1)d$ $26 = 8 + 3d$

Subtract 8 from both sides: $26 - 8 = 3d$ $18 = 3d$

Divide by 3: $d = \frac{18}{3} = 6$

Step 2: Find $p$ $p$ is the second term ($T_2$): $p = a + d$ $p = 8 + 6$ $p = 14$

Step 3: Find $q$ $q$ is the third term ($T_3$): $q = p + d$ $q = 14 + 6$ $q = 20$

Conclusion

We have calculated $p = 14$ and $q = 20$. Comparing this to the given options, it matches option D.

Exam Tip

In an A.P. with four terms, you can also use the property that the sum of the extremes equals the sum of the means ($T_1 + T_4 = T_2 + T_3$). Here, $8 + 26 = 34$, and $14 + 20 = 34$. This is a quick way to verify your answer during the exam!