Use mathematical induction to prove that the statement is true for every positive integer n.4 ? 6 + 5 ? 7 + 6 ? 8 + . . . +(n + 3)(n + 5) = 

What will be an ideal response?


Answers will vary. One possible proof follows.
a). Let n = 1. Then 4 ? 6 = 24 =  =  = 24. Thus, the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 Sk = 
 Also, if the statement is true for n = k + 1, then
 Sk+1 = Sk + ((k + 1) + 3)((k + 1) + 5) = 
 Subtract to get:
 Sk+1 - Sk  = ((k + 1) + 3)((k + 1) + 5) =  - 
 Expand both sides and collect like terms:
 k2 + 10k + 24 =  -  = 
 k2 + 10k + 24 = k2 + 10k + 24
Since the equality holds, then the statement is true for n = k + 1 as long as it is true for n = k. Furthermore, it is true for n = 1. Therefore, the statement is true for all natural numbers n.

Mathematics

You might also like to view...

Simplify the rational expression. If the rational expression cannot be simplified, so state.

A. x - y
B. cannot be simplified
C.
D. x - 2y + 1

Mathematics

Find the exact value of the expression.sin 

A.
B.
C.
D.

Mathematics

Indicate whether the graph of the equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. +  = 1

A. parabola

B. ellipse

C. hyperbola

D. circle

Mathematics

Give the reciprocal of the given number. ? 

A.
B.
C.
D.
E.

Mathematics