What is common difference of an AP., in which ${{\text{a}}_{{\text{21}}}}{\text{ - }}{{\text{a}}_{\text{7}}}{\text{ = 644}}$ .

Hint: Arithmetic progression is the sequence of terms in which the successive difference between any two consecutive terms is the same.
Formulae Required: ${{\text{a}}_{\text{n}}}{\text{ = a + }}\left( {{\text{n - 1}}} \right){\text{d}}$
Where, ${{\text{a}}_{\text{n}}}$ is the nth term in the arithmetic progression
${\text{a}}$ is the first term in the arithmetic progression
${\text{n}}$ is the number of terms in the arithmetic progression
${\text{d}}$ is the common difference between any two consecutive terms in the arithmetic progression

Complete step-by-step answer:
Given: ${{\text{a}}_{{\text{21}}}}{\text{ - }}{{\text{a}}_{\text{7}}}{\text{ = 644}}$
Where, ${{\text{a}}_{{\text{21}}}}$ is the 21st term in the arithmetic progression
${{\text{a}}_{\text{7}}}$ is the 7th term in the arithmetic progression
We need to find the common difference (${\text{d}}$)
According to the question ,
${{\text{a}}_{{\text{21}}}}{\text{ - }}{{\text{a}}_{\text{7}}}{\text{ = 644}}$ …………..equation (1)
Now applying the formula of nth term in the arithmetic progression i.e, ${{\text{a}}_{\text{n}}}{\text{ = a + }}\left( {{\text{n - 1}}} \right){\text{d}}$ we get
$
   \Rightarrow {{\text{a}}_{{\text{21}}}}{\text{ = a + }}\left( {{\text{21 - 1}}} \right){\text{d}} \\
   \Rightarrow {{\text{a}}_{{\text{21}}}}{\text{ = a + }}\left( {{\text{20}}} \right){\text{d}} \;
$
Similarly,
$
   \Rightarrow {{\text{a}}_7}{\text{ = a + }}\left( {{\text{7 - 1}}} \right){\text{d}} \\
   \Rightarrow {{\text{a}}_7}{\text{ = a + }}\left( 6 \right){\text{d}} \;
$
From equation (1) we have,
${{\text{a}}_{{\text{21}}}}{\text{ - }}{{\text{a}}_{\text{7}}}{\text{ = 644}}$
$
   \Rightarrow \left( {{\text{a + 20d}}} \right){\text{ - }}\left( {{\text{a + 6d}}} \right){\text{ = 644}} \\
   \Rightarrow {\text{20d - 6d = 644}} \\
   \Rightarrow {\text{14d = 644}} \\
   \Rightarrow {\text{d = }}\dfrac{{{\text{644}}}}{{{\text{14}}}} \\
   \Rightarrow {\text{d = 46}} \;
$
Therefore the common difference (${\text{d}}$) is 46.
So, the correct answer is “46”.

Note: In this type of questions which involves the concept of arithmetic progression, knowledge about the formula related to arithmetic progression is necessary. We will need to be vigilant with the calculations involved.