Question

The sum of the reciprocals of two consecutive integers is 3/2
. Find the two integers.

Answer 1

The two consecutive integers whose reciprocals sum up to 3/2 are 1 and 2, found by solving the equation 1/n + 1/(n+1) = 3/2, with n representing the smaller integer.

Step-by-step explanation:

The sum of the reciprocals of two consecutive integers is 3/2. To find these integers, let's denote the smaller integer as n, so the next consecutive integer is n + 1. The equation based on the given sum of reciprocals is therefore 1/n + 1/(n + 1) = 3/2.

To solve this, we need a common denominator which is n(n + 1). This transforms the equation into (n + 1) + n = 3n(n + 1)/2. By simplifying and solving for n, we find that the two consecutive integers are 2 and 3.

Here is the step-by-step process:

1. 
   
3. Write the equation with a common denominator: (n + 1)/n(n + 1) + n/n(n + 1) = 3/2.
4. 
   
6. Simplify the equation: 2(n + n + 1) = 3n(n + 1).
7. 
   
9. Expand and rearrange: 2n + 2 = 3n + 3.
10. 
    
12. Solve for n: 2 = n + 3; n = -1, which is not possible for positive integers.
13. 
    
15. Since we are looking for positive integers, let's verify again and notice that the intended steps might include a mistake, so we retry by correctly expanding: 2(2n + 1) = 3n(n + 1).
16. 
    
18. 4n + 2 = 3n^2 + 3n.
19. 
    
21. Rearrange to form a quadratic equation: 0 = 3n^2 - n - 2.
22. 
    
24. Factor the quadratic: (3n + 2)(n - 1) = 0.
25. 
    
27. Solve for n: n = -2/3 (discarded as we need an integer) or n = 1.
28. 
    
30. The two consecutive integers are 1 and 2, which indeed have the reciprocals summing up to 3/2.