Solve the question 4 1 2 5 x 437

Asked by admin @ 23/08/2021 in Math viewed by 431 People

Solve the equation ( - 4 ) + ( - 1 ) + 2 + .... + x = 437.

Answered by admin @ 23/08/2021

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

$\textsf{The above series forms an A.P.}$

$\mathsf{Where, } \\ \\\\\mathsf{ \implies First \: term(a) \: = \: -4 } \\ \\\\\mathsf{\implies Last \: term (l) \: = \: x } \\ \\\\\mathsf{\implies Common \: difference (d) \: = \: (-1) \: - \: (-4)} \\ \\ \mathsf{ \qquad \qquad \qquad \qquad \qquad \qquad \: = \: -1 \: + \: 4 }\: \\ \\ \mathsf{ \qquad \qquad \qquad \qquad \qquad \qquad \: = \: 3 }$

$\mathsf{Let, no. \: of \: terms \: is \: n. }$

$\mathsf{Using \: Formula, } \\ \\\\\mathsf{ \implies l \: = \: a \: + \: (n \: - \: 1)d } \\ \\\\\mathsf{\implies x \: = \: (-4) \: + \: (n \: - \: 1)3 } \\ \\\\\mathsf{\implies x \: = \: -4 \: + \: 3n \: - \: 3 } \\ \\\\\mathsf{\implies x\: = \: 3n \: - \: 7 } \\ \\\\\mathsf{\implies x\: + \: 7 \: = \: 3n } \\ \\\\\mathsf{\therefore \quad n \: = \: \dfrac{( x \: + \: 7)}{3}}$

$\underline{\mathsf{Now, }} \\ \\\\\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}( a \: + \: l )} \\ \\\textsf{Plug the value of \textbf{n}, } \\ \\ \\ \\\mathsf{\implies 437 \: = \: \dfrac{ \quad(\dfrac{x\: + \: 7}{3}) \quad}{2}( -4 \: + \: x )}$

$\\ \mathsf{\implies 437 \: \times \: 2 \: = \: \dfrac{(x \: + \: 7)}{3}(x \: - \: 4 )} \\ \\\\\mathsf{\implies 437 \times \: 2 \: \times \: 3 \: = \: (x \: + \: 7 )( x \: - \: 4 )}$

$\\ \mathsf{\implies 2622 \: = \: {x}^{2} \: - \: 4x \: + \: 7x \: - \: 28 } \\ \\ \mathsf{ \implies {x}^{2} \: + \: 3x \: - \: 28 \: - \: 2622 \: = \: 0 } \\ \\ \mathsf{ \implies {x}^{2} \: + \: 3x \: - \: 2650 \: = \: 0}$

$\mathsf{Here, } \\ \\\\\mathsf{\implies Coefficient \: of \: {x}^{2}(a) \: = \: 1 } \\ \\\\\mathsf{\implies Coefficient \: of \: x (b ) \: = \: 3 } \\ \\\\\mathsf{\implies Constant \: term(c) \: = \: -2650 }$

$\textsf{Using Quadratic Formula , } \\ \\ \mathsf{ \implies x \: = \: \dfrac{ - b \: \pm \: \sqrt{ \: {b}^{2} \: - \: 4ac \: } }{2a} }$

$\\ \mathsf{ \implies x \: = \: \dfrac{ - 3 \: \pm \: \sqrt{ \: ( {3}^{2} ) \: - \: 4 \: \times \: 1 \: \times \: ( - 2650)} }{2 \: \times \: 1} } \\ \\ \\ \mathsf{ \implies x \: = \: \dfrac{ - 3 \: \pm \: \sqrt{ \: 9 \: + \: 10600} }{2}}$

$\\ \mathsf{ \implies x \: = \: \dfrac{ - 3 \: \pm \: \sqrt{ \: 10609} }{2} } \\ \\ \\ \mathsf{ \implies x \: = \: \dfrac{ - 3 \: \pm \: 103}{2}}$

$\\ \mathsf{\implies x \: = \: \dfrac{-3 \: + \: 103}{2} \quad, \implies x \: = \: \dfrac{-3 \: - \: 103}{2}} \\ \\ \\\\\mathsf{\implies x \: = \: \dfrac{100}{2} \quad, \implies x \: = \: \dfrac{-106}{2}} \\ \\ \\\\\mathsf{\implies x \: = \: 50 \: or \: (-53 ) \: [ Not \: Possible ]}$

$\boxed{\underline{\mathsf{Therefore, x \: is \: equal \: to \: 50.}}}$

Solve the question 4 1 2 5 x 437

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