Does there exist a pair of consecutive even integers whose reciprocals sum to 3/4? Explain how you know.

Accepted Solution

Answer:2, 4Step-by-step explanation:Let there are two unknown consecutive pairs of even integers 2n and 2n+2. If the sum of the reciprocals of those two integers Β is 3/4, then we can write, 1/2n +1/(2n+2) =3/4 ...... (1) Solving this above equation we can check whether the integers exist or not. Now, from equation (1) we get, 1/n +1/(n+1) =3/2 β‡’[tex]2(2n+1)=3n(n+1)[/tex] β‡’4n +2 =3nΒ²+3n β‡’3nΒ²-n-2 =0 β‡’ (n-1)(3n+2) =0 β‡’ (n-1) =0 {Since if (3n+2) =0 then, n will be a negative fraction, which is not possible} β‡’ n=1 ∴ 2n =2 and 2n+2 =4 Hence, the two integers are 2, 4. (Answer)