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Relation between AM,GM,HM for 2 Numbers

a 1, a 2, a 3...

Question

a1,a2,a3,a4,a5 are the first five terms of an ap such that a1+a3+a5= -12 & a1a2a3=8 find the first term and common difference

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Solution

Let d be the common difference .

A1+ (A1+2d) + (A1+4d) = -12 and

A1(A1+2d) (A1+4d) = 8 and

solve equations for A1 and d

Now A2+A4+A6 = (A1+d) + (A1+3d) + (A1+5d).

Substitute values for A1 and d found in previous step.

Shorter method

Let d be the common difference .

(A3-2d)+ A3 + (A3+2d) = -12 and

A3 = -4

(A3-2d)* (A3-d) *(A3) = 8

Solve for d

Now A2+A4+A6 = (A3-d) + (A3+d) + (A3+2d).

Substitute values for A3 and d found in previous step.

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Similar questions

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

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

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

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

a_{1} = 0

d \neq 0

\frac{a_{3} - a_{2}}{a_{2}} + \frac{a_{4} - a_{2}}{a_{3}} + \frac{a_{5} - a_{2}}{a_{4}} . . . . . . . . . . . . \frac{a_{n} - a_{2}}{a_{n - 1}}

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

n

a = 0

d \neq 0

\frac{a_{3} - a_{2}}{a_{2}} + \frac{a_{4} - a_{2}}{a_{3}} + \frac{a_{5} - a_{2}}{a_{4}} . . . . + \frac{a_{n} - a_{2}}{a_{n - 1}}

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

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

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

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

A . P .

a_{1}, a_{2}, a_{3}, . . . . ., a_{n}, . . . ., a_{1} + a_{3} + a_{5} = - 12

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

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

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