Question

find the term of the arithmetic sequence whose sum of the reciprocal of first two consecutive terms is 1/6 and common difference is 5?​

Answer 1

Answer:

Let the first term of the AP be x

and the second term of the AP be y.

According to the question,

The sum of the reciprocals of first two terms is 1/6.

=> 1/x + 1/y = 1/6

=> (y+x)/xy = 1/6

=> 6(x+y) = xy --------(1)

Also,

Common difference is 5;

=> y - x = 5

=> y = x + 5 ------(2)

Now, From eq-(1) and (2) ,we have;

=> 6(x+y) = xy

=> 6(x+x+5) = x(x+5)

=> 6(2x+5) = x(x+5)

=> 12x + 30 = x^2 + 5x

=> x^2 + 5x -12x - 30

=> x^2 - 7x - 30 = 0

=> x^2 - 10x + 3x - 30 = 0

=> x(x - 10) + 3(x - 10) = 0

=> (x-10)(x+3) = 0

=> x = 10 or x = -3

Now;

If x = 10,

Then, y = x + 5. {using eq-(2)}

=> y = 10 + 5

=> y = 15

If x = -3

Then, y = x + 5. {using eq-(2)}

=> y = -3 + 5

=> y = 2

Hence, two APs are possible ;

First AP;

First term = 10

Second term = 15

Second AP;

First term = -3

Second term = 2