Question

a1 ,a2 ,a3, a4 ,a5 are first five terms of an AP . Such that a1 +a2+a3 =-12 and a1 a2a3 =8 . Find the first term and difference

Answer 1

Hope it helped you .

Answer 2

$Let \: a_{1} = (a-d), a_{2} = a , a_{3} = a+d, \\a_{4} = a+2d, \: and \: a_{5} = a+3d \: are \: first \\five \: terms \: of \: an \: A.P$

$a_{1} + a_{2} + a_{3} = -12 \: (given)$

$\implies (a-d) + a + (a+d) = -12$

$\implies a-d + a + a+d= -12$

$\implies 3a = -12$

$\implies a = \frac{-12}{3}$

$\implies a = - 4 \: --(1)$

$a_{1} a_{2} a_{3} = 8 \: (given)$

$\implies (a-d)a (a+d) = 8$

$\implies (a^{2}-d^{2})a = 8$

$\implies [ (-4)^{2} - d^{2} ] (-4) = 8$

$\implies [ (-4)^{2} - d^{2} ] = \frac{8}{-4}$

$\implies 16 - d^{2} = -2$

$\implies - d^{2} = -2 - 16$

$\implies - d^{2} = - 18$

$\implies d^{2} = 18$

$\implies d = \pm \sqrt{18}$

$\implies d = \pm 3\sqrt{2}\: ---(2)$

Case 1:

/* If a = -4, and d = 3√2 */

$First \:term \: in \: A.P = a - d \\= -4 - 3\sqrt{2}$

$\red{Common \: difference (d)} = \green {3\sqrt{2}}$

Case 2:

/* If a = -4, and d = -3√2 */

$First \:term \: in \: A.P = a - d \\= -4 - (-3\sqrt{2})\\= -4 + 3\sqrt{2}$

$\red{Common \: difference (d)} = \green {-3\sqrt{2}}$

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