Use mathematical induction to prove that the statement is true for every positive integer n.If 0 < a < 1, then an < 1.(Assume that a is a constant.)

What will be an ideal response?


Answers will vary. One possible proof follows.
a). Let n = 1. Then, a1 < 1 or a < 1, which is true by assumption.
b). Assume that the statement is true for n = k:
 ak < 1
 Multiply both sides by a:
 a?ak = ak+1 < a
By assumption, a < 1, so ak+1 < 1. Since the statement is true for n = k + 1 as long as it is true for n = k, and since the statement is true for n = 1, then it is true for all natural numbers n.

Mathematics

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A.  log17mn -  log178
B.  log17m +  log17n - log178
C.  log17m ?  log17n ÷  log178
D.  log17m +  log17n -  log178

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Given points P, Q, R, and S, determine whether lines  and are perpendicular, parallel, or neither.P(1, 4), Q(5, 8), R(3, 0), S(15, -12)

A. Neither B. Perpendicular C. Parallel

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Use a calculator to perform the indicated operation. Round the result to two decimal places.-  

A. 5.40 B. 0.19 C. -5.40 D. -0.19

Mathematics

Solve the problem. Express your answer in scientific notation, rounding as needed.The earth is approximately 92,900,000 miles from the sun.  × 103 m, what is the distance to the sun in meters?

A. 5.7 × 10-10 m B. 1.50 × 1011 m C. 5.7 × 1010 m D. 1.50 × 1010 m

Mathematics