27 ### Educational Assessment Questions**21.** True or false? If \( m \) is any even integer and \( n \) is any odd integer, then \( m^2 + 3n \) is odd. Explain.**22.** True or false? If \( a \) is any odd integer, then \( a^2 + a \) is even. Explain.**23.** True or false? If \( k \) is any even integer and \( m \) is any odd integer, then \( (k+2)^2 - (m-1)^2 \) is even. Explain.### Derived StatementsDerive the statements in questions 24–26 as corollaries of Theorems 4.3.1, 4.3.2, and the results of exercises 12, 13, 14, 15, and 17.**24.** For any rational numbers \( r \) and \( s \), \( 2r + 3s \) is rational.**25.** If \( r \) is any rational number, then \( 3r^2 - 2r + 4 \) is rational.**26.** For any rational number \( s \), \( 5s^3 + 8s^2 - 7 \) is rational.**27.** It is a fact that if \( n \) is any nonnegative integer, then\[1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots + \frac{1}{2^n} = \frac{1 - (1/2^{n+1})}{1 - (1/2)}\](A more general form of this statement is proved in Section 5.2.) Is the right-hand side of this equation rational? If so, express it as a ratio of two integers.**28.** Suppose \( a, b, c, \) and \( d \) are integers and \( a \neq c \). Suppose also that \( x \) is a real number that satisfies the equation\[\frac{ax + b}{cx + d} = 1.\]

Answered: Image /qna-images/answer/bbe070ec-1647-4400-9fbb-a0ed2803f61f.jpg

It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2) 1 2" (A more general form of this statement is proved in Section 5.2.) Is the right-hand side of this equation rational? If so, express it as a ratio of two integers.