Question

sum of three number of an ap is 27 and their product is 460 find the AP ​

Answer 1

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: ANSWER \: \: } \mid}}}}}$

$\text{Suppose,} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{The \: three \: numbers \: of \: the \: AP \:} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \text{ are \: (a-d), a \: and \: (a+d).} \\ \\ \text{Where,} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{ \red{a} \: \: is \: \: the \: \: first \: \: term \: \: and \: \: \red{d} \: \: is \: } \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{ the \: \: difference.} \\ \\ </p><p>$

$\underline{ \bold{ \: Given \: : \mapsto}} \\ \\ \: \: \: \: \: \text{Sum \: of \: the \: three \: number = 27} \\ \\ \mathsf{ \therefore \: (a-d)+a+(a+d) = 27} \\ \\ \mathsf{ \Rightarrow \: a - d + a + a + d = 27} \\ \\ \mathsf{ \Rightarrow \: 3a = 27} \\ \\ \mathsf{ \Rightarrow \: a = \frac{27}{3} } \\ \\ \bold{ \therefore \: \: \underline{ \: a = 9 \: }} \\ \\ </p><p></p><p>$

$\underline{ \bold {\: \: Again \: : \mapsto}} \\ \\ \text{Product \: of \: the \: three \: numbers = 460} \\ \\ \mathsf{ \therefore \: (a-d) \times a \times (a+d) = 460} \\ \\ \mathsf{ \Rightarrow \: (a-d) \times (a+d) \times a = 460} \\ \\ \mathsf{ \Rightarrow \: (a {}^{2} -d {}^{2} ) \times a = 460} \\ \\ \mathsf{ \Rightarrow \: (9 {}^{2} -d {}^{2} ) \times 9= 460} \\ \\ \mathsf{\Rightarrow \: (81 - d {}^{2} ) \times 9 = 460} \\ \\ \mathsf{\Rightarrow \: 81 - d {}^{2} = \frac{460}{9} } \\ \\ \mathsf{\Rightarrow \: 81 - d {}^{2} = 51.11} \\ \\ \mathsf{ \Rightarrow \: - d {}^{2} = 51.11 - 81 } \\ \\ \mathsf{\Rightarrow \: - d {}^{2} = - 29.89 } \\ \\ \bold{ \therefore \: \: \underline{ \: d = \pm \: 5.46 \: }}$

$\text{If \: \: d = + 5.46}, \\ \\ \bold{ \therefore \: \: a - d = 9 -5.46 = 3.54} \\ \\ \: \: \: \: \: \: \: \bold{a + d = 9 + 5.46 = 14.46} \\ \\ \text{Then, } \\ \: \: \: \: \: \: \: \text{The \: AP \: will \: be \: : \: 3.54 , 9 , 14.46 }$

$\text{If \: \: d = - 5.46}, \\ \\ \bold{ \therefore \: \: a - d = 9 - ( -5.46) = 14.46} \\ \\ \: \: \: \: \: \: \: \bold{a + d = 9 +( - 5.46) = 3.54} \\ \\ \text{Then, } \\ \: \: \: \: \: \: \: \text{The \: AP \: will \: be \: : \: 14.46 , 9 , 3.54}$