Question

❤️❤️❤️❤️HELLO MY FRIENDS AND BRAINLY STARS❤️❤️❤️❤️

please answer my question.

I will mark as brainlist.................

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Answer 1

hey mate
here's the solution

Answer 2

$\underline{\underline{\large{\mathfrak{Solution : }}}}$



$\textsf{Let,} \\ \\ \mathsf{\implies First \: term \: = \: a } \\ \\ \mathsf{\implies Common \: difference \: = \: d }$




$\underline{\textsf{According to question : }} \\ \\ \mathsf{\implies T_{ \normalsize{(m \: + \: 1 )}} \: = \: 2 \: \times \: T_{ \normalsize{(n \: + \: 1)}}}$




$\textsf{Using Formula : } \\ \\ \boxed{\mathsf{\implies T_{n} \: = \: a \: + \: ( n \: - \: 1)d }}$




$\mathsf{ \implies a \: + \: ( m \: + \: \cancel{1 }\: - \: \cancel{1})d \: = 2 \: \times \{ \: a \: + \: ( \: n \: + \: \cancel{ 1} \: - \: \cancel{1})d \}} \\ \\ \mathsf{ \implies a \: + \: md \: = \: 2(a \: + \: nd)} \\ \\ \mathsf{ \implies a \: + \: md \: = \: 2a \: + \: 2nd } \\ \\ \mathsf{ \implies md \: - \: 2nd \: = \: 2a \: - \: a } \\ \\ \mathsf{ \therefore \quad \: a \: = \: md \: - \: 2nd }$




$\textsf{Now,}$



$\mathsf{\implies T_{\normalsize{(3m \: + \: 1 })} \: = \: a \: + \: ( 3m \: + \: \cancel{1} \: - \: \cancel{1 })d }$



$\mathsf{\implies T_{\normalsize{( 3m \: + \: 1 )}} \: = \: a \: + \: 3md }$




$\textsf{Plug the value of \textbf{a} : }$




$\mathsf{\implies T_{\normalsize{(3m \: + \: 1)}} \: = \: md \: - \: 2nd \: + \: 3md } \\ \\ \mathsf{\implies T_{\normalsize{(3m \: + \: 1)}} \: = \: 4md \: - \: 2nd } \\ \\ \mathsf{\implies T_{\normalsize{(3m \: + \: 1)}} \: = \: 2(2md \: - \: nd) \qquad...(1) }$



$\textsf{Again, } \\ \\ \mathsf{\implies T_{\normalsize{(m \: + \: n \: + \: 1)}} \: = \: a \: + \: ( m \: + \: n \: + \:\cancel{ 1} \: - \: \cancel{1 })d} \\ \\ \\ \textsf{Plug the value of \textbf{a} : } \\ \\ \mathsf{\implies T_{\normalsize{(m \: + \: n \: + \: 1)}} \: = \: md \: - \: 2nd \: + \: md \: + \: nd } \\ \\ \mathsf{\implies T_{\normalsize{(m \: + \: n \: + \: 1)}} \: = \: 2md \: - \: nd \qquad...(2)}$




$\textsf{Plug the value of (2) in ( 1), } \\ \\ \mathsf{\implies T_{\normalsize{(3m \: + \: 1)}} \: = \: 2 \: \times \: T_{\normalsize{(m \: + \: n \: + \: 1)}} }$




$\boxed{\underline{\large{\mathsf{Proved \: !! }}}}$