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:

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

  2. 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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