Question

The sum of the reciprocals of 2 consecutive integers is 19/90 . Find the numbers.

Answer 1

Step-by-step explanation:

we consider the two consecutive numbers as x and x+1

Answer 2

The consecutive integers are 9 and 10, whose sum of reciprocals is 19/90.

Given:

• Sum of reciprocals of 2 consecutive integers is 19/90.

To find:

• Find the numbers.

Solution:

Concept to be used:

• Assume to consecutive integers as variables.
• Write the equation and solve.
• Quadratic formula $\bf x_{1,2} = \frac{-b \pm \sqrt{( {b}^{2} - 4ac} }{2a}$

Step 1:

Write the equation.

Let 'x' and 'x+1' are two consecutive integers.

So, ATQ

$\frac{1}{x} + \frac{1}{x + 1} = \frac{19}{90}$

Simplify the equation.

$\frac{x + 1 + x}{x(x + 1)} = \frac{19}{90}$

$(2x + 1)90 = 19( {x}^{2} + x)$

or

$19 {x}^{2} + 19x = 180x + 90$

$\bf 19 {x}^{2} - 161x - 90 = 0$

Step 2:

Solve the quadratic equation.

Apply quadratic formula.

$x_{1,2} = \frac{161 \pm \sqrt{( {161)}^{2} - 4(19)( - 90)} }{19 \times 2}$

$x_{1,2} = \frac{161 \pm \sqrt{25921 + 6840} }{38}$

$x_{1,2} = \frac{161 \pm 181 }{38}$

Take (+)ve sign.

$x_1 = \frac{161 + 181}{38}$

$x_1 = \frac{342}{38}$

$\bf x_1 = 9$

So, The first number is 9 and the second is 10.

Take (-)ve sign.

$x_2 = \frac{161 - 181}{38}$

$x_2 = \frac{ - 20}{38}$

$\bf \red{x_2 = \frac{ - 10}{19} }$

We have to discard this value, because it is not an integer.

Thus,

The consecutive integers are 9 and 10, whose sum of reciprocals is 19/90.

Learn more:

1) If the sum of three consecutive numbers is 42, find these numbers.

https://brainly.in/question/5039499

2) The sum of two numbers is 22 and the sum of their squares is 250. Find the numbers.

brainly.in/question/27417573

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