Find the common difference of the given AP: 3/4, 1, 1 1/4 ... ... A) 1/4 B) 1/4 C) 1/3 D) 1/2
Question
Find the common difference of the given AP: $\frac{3}{4}, 1, 1 \frac{1}{4} \ldots \ldots $.
A) $\frac{1}{4}$ B) $\frac{1}{4}$ C) $\frac{1}{3}$ D) $\frac{1}{2}$
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Answer
:
The correct option is A: $$ \frac{1}{4} $$
When a series is in Arithmetic Progression (A.P.), the difference between consecutive terms, known as the common difference, remains constant.
Given the sequence: $$ \frac{3}{4},\ 1,\ 1 \frac{1}{4}, \ldots $$
Let's identify the terms:
 First term, $a_1 = \frac{3}{4}$
 Second term, $a_2 = 1$
 Third term, $a_3 = 1 \frac{1}{4}$
To find the common difference $d$, we calculate the difference between consecutive terms:

Between the first and second term: $$ d = a_2  a_1 = 1  \frac{3}{4} = \frac{4}{4}  \frac{3}{4} = \frac{1}{4} $$

Between the second and third term: $$ d = a_3  a_2 = 1 \frac{1}{4}  1 = \frac{5}{4}  1 = \frac{5}{4}  \frac{4}{4} = \frac{1}{4} $$
Since the common difference $d$ is consistently $\frac{1}{4}$ between each pair of consecutive terms, the common difference of the given A.P. is:
$$ \boxed{\frac{1}{4}} $$
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