Question

The sum of three number in AP is minus 3 and their product is 8. Find the number

Answer 1

QUESTION➡The sum of three number in AP is minus 3 and their product is 8. Find the number

TO FIND➡first term & common difference?

GIVEN➡1)sum of 3 number=-3

2)product of 3 number=8

$\bf\red{Let \: the \: three \: numbers \: in \: AP \: are \mapsto} \\ \\ \huge\boxed{ \pink{ a-d, a, a + d}} \\ \\ \bf\green{ where \: d \: is \: the \: common \: difference \:} \\ \\ \bf\ their \: sum = - 3 \\ \\ \bf\ \therefore \: a - d + a + a + d = - 3 \\ \\ \bf\ \implies: 3a = - 3 \\ \\ \bf\ \implies: a = \frac{ - 3}{3} \\ \\ \bf\boxed{ \implies: a = - 1} \\ \\ \bf\ \: their \: product = 8 \\ \\ \bf\ \implies: (a + d) \times a \times (a - d) = 8 \\ \\ \bf\ \implies: a( {a }^{2} - {d}^{2} ) = 8 \\ \\ \bf\ \implies: ( - 1)( { - 1}^{2} - {a}^{2} ) = 8 \\ \\ \bf\ \implies: - 1 + {d}^{2} = 8 \\ \\ \bf\ \implies: {d}^{2} = 9 \\ \\ \bf\ \implies: d = ±\sqrt{9} \\ \\ \bf\boxed{ \implies: d = ±3} \\ \\ \huge\purple{\boxed{ \mid{ \underline{ \overline{ a = - 1 \: and \: d = ±3}}}}} \\ \\ \bf\orange{when \: d = 3 \: numbers \: are \leadsto} \\ \\ \bf\orange{a + d = -1 + 3 = - 2 , \: a = - 1 , \: a - d = - 1 - 3 = - 1 - 3 = - 4} \\ \bf\orange{ - 2, - 1, 2} \\ \\ \\ \bf\blue{when \: d = - 3 \: numbers \: are \leadsto} \\ \\ \bf\blue{a + d = -1 + ( - 3) = - - 4 , \: a = - 1 , \: a - d = - 1 - ( - 3) = - 1 + 3 = - 2} \\ \bf\blue{ - 4, - 1, - 2} \\ \\ \huge\purple{\mathtt{hope \: it \: help \: you} }$