For an increasing A.P. $a_{1}, a_{2}, a_{3}, \dots, a_{n},$ if $a_{1} + a_{3} + a_{5} = - 12$
and $a_{1} a_{3} a_{5} = 80$ , then which of the following is (are) TRUE?

a_{1} = - 10

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a_{2} = - 1

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a_{3} = - 4

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a_{5} = 2

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Solution

The correct options are
A $a_{1} = - 10$
C $a_{3} = - 4$
D $a_{5} = 2$
Let $a_{1} = a$ and $d$ be the common difference.
Given, $a_{1} + a_{3} + a_{5} = - 12$
$\Rightarrow a + (a + 2 d) + (a + 4 d) = - 12 \Rightarrow 3 a + 6 d = - 12 \Rightarrow a + 2 d = - 4 \dots (1)$

and $a_{1} a_{3} a_{5} = 80$
$\Rightarrow a (a + 2 d) (a + 4 d) = 80$
$\Rightarrow (- 4 - 2 d) (- 4) (- 4 - 2 d + 4 d) = 80 [From (1)]$
$\Rightarrow (- 4 - 2 d) (- 4 + 2 d) = - 20 \Rightarrow (2 d)^{2} - 4^{2} = 20 \Rightarrow 4 d^{2} = 36 \Rightarrow d = \pm 3$

Since A.P. is increasing, so $d = + 3.$
From eq $(1)$ , $a = - 10$
Hence, $a_{1} = - 10$
$a_{2} = a + d = - 7 a_{3} = a + 2 d = - 4 a_{5} = a + 4 d = 2$

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Similar questions

For an increasing A.P. $a_{1}, a_{2}, a_{3} . . . a_{n}$ ,
if $a_{1} + a_{3} + a_{5} = - 12$ and $a_{1} \cdot a_{3} \cdot a_{5} = 80$ , then which of the following is true.

Q. If

a_{1}, a_{2}, a_{3}, a_{4}, a_{5} > 0

then

\frac{a_{1}}{a_{2} + a_{3}} + \frac{a_{2}}{a_{3} + a_{4}} + \frac{a_{3}}{a_{4} + a_{5}} + \frac{a_{4}}{a_{5} + a_{1}} + \frac{a_{5}}{a_{1} + a_{2}} \geq \frac{5}{2}

Q. If for an

A . P .

a_{1}, a_{2}, a_{3}, . . . . . ., a_{n}, . . . . .

a_{1} + a_{3} + a_{5} = - 12

and

a_{1} a_{2} a_{3} = 8

then the value of

a_{2} + a_{4} + a_{6}

equals

Q. If for an A.P.

a_{1}, a_{2}, a_{3} . . . . . . . . . a_{n},

a_{1} + a_{3} + a_{5} = - 12

and

a_{1} a_{2} a_{3} = 8

then the value of

a_{2} + a_{4} + a_{6}

equals

Q. If

a_{1}, a_{2}, a_{3}, . . . . . . . . . ., a_{n}

are consecutive terms of an increasing

A . P .

and

(1^{2} - a_{1}) + (2^{2} - a_{2}) + (3^{2} - a_{3}) + . . . . . . . + (n^{2} - a_{n}) = \frac{(n - 1) n (n + 1)}{3}

, then the value of

(\frac{a_{5} + a_{3} - a_{2}}{6})

is equal to

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For an increasing A.P. a1,a2,a3,…,an, if a1+a3+a5=−12 and a1a3a5=80, then which of the following is (are) TRUE?

For an increasing A.P. $a_{1}, a_{2}, a_{3}, \dots, a_{n},$ if $a_{1} + a_{3} + a_{5} = - 12$
and $a_{1} a_{3} a_{5} = 80$ , then which of the following is (are) TRUE?