# Math

2. The sum of the reciprocals of two consecutive positive integers is 17/72. Write an equation that can be used to find the two integers. What are the integers?

Steve helped me yesterday and gave me the hint 8+9=17. Then I thought about it and saw 8*9=72, but I am confused about how to put it into an equation.

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1. x = first number

x + 1 = second number

1 / x + 1 / ( x + 1 ) = 17 / 72

[ 1 * ( x + 1 ) + 1 * x ] / [ x * ( x + 1 ) ] = 17 / 72

[ ( x + 1 ) + x ] / [ x * x + x * 1 ) ] = 17 / 72

( 2 x + 1 ) / ( x ^ 2 + x ) = 17 / 72 Multiply both sides by 72

72 * ( 2 x + 1 ) / ( x ^ 2 + x ) = 17

( 72 * 2 x + 72 * 1 ) / ( x ^ 2 + x ) = 17

( 144 x + 72 ) / ( x ^ 2 + x ) = 17 Multiply both sides by x ^ 2 + x

144 x + 72 = 17 * ( x ^ 2 + x )

144 x + 72 = 17 * x ^ 2 + 17 * x

144 x + 72 = 17 x ^ 2 + 17 x Subtract 144 x to both sides

144 x + 72 - 144 x = 17 x ^ 2 + 17 x - 144 x

72 = 17 x ^ 2 - 127 x Subtract 72 to both sides

72 - 72 = 17 x ^ 2 - 127 x - 72

0 = 17 x ^ 2 - 127 x - 72

17 x ^ 2 - 127 x - 72 = 0

The solutions are:

x = - 9 / 17 and x = 8

- 9 / 17 isn't positive integers so x = 8

first number = 8

second number = 8 + 1 = 9

Proof:

1 / x + 1 / ( x + 1 ) =

1 / 8 + 1 / 9 =

( 1 * 9 + 1 * 8 ) / ( 8 * 9 ) =

( 9 + 8 ) / 72 = 17 / 72

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