Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
What will be an ideal response?
When 
, the left side of the statement is 
, and the right side of the
statement is 
, so the statement is true when 
.
Assume the statement is true for some natural number k. Then,
.
So the statement is true for 
. Conditions I and II are satisfied; by the Principle of Mathematical Induction, the statement is true for all natural numbers.
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Write a system of three inequalities that describe the constraints in the problem.An office manager is buying used filing cabinets. Small file cabinets cost $4 each and large file cabinets cost $11 each, and the manager cannot spend more than $57 on file cabinets. A small cabinet takes up 6 square feet of floor space and a large cabinet takes up 9 square feet, and the office has no more than 63 square feet of floor space available for file cabinets. The manager must buy at least 6 file cabinets in order to get free delivery. Let x = the number of small file cabinets bought and y = the number of large file cabinets bought.
A. 4x + 11y ? 57; 9x + 6y ? 63; x ? 6 B. 4x + 11y ? 57; 6x + 9y ? 63; x + y ? 6 C. 4x + 11y ? 57; 6x + 9y ? 63; x + y ? 6 D. 4x + 11y ? 57; 6x + 9y ? 63; y ? 6
In the given voting system, the weights represent, in order, voters A, B, C, and so on. Determine which voters are critical in the indicated coalition.[23 : 3, 5, 6, 8, 9] ; coalition {A, B, D, E}
A. Only A and B are critical. B. Only D and E are critical. C. None are critical. D. All are critical.
Factor the binomial completely.49a3 - 25a
A. a(49a + 1)(a - 25) B. (7a2 + 5)(7a - 5) C. a(7a + 5)(7a - 5) D. a(7a - 5)2
Set up the percent proportion for the application problem. Do not try to solve for any unknowns.10% of Tom's check of $500 is withheld. How much is withheld?
A. 
= 
B. 
= 
C. 
= 
D. 
= 