Use mathematical induction to prove the statement is true for all positive integers n.8n > n
What will be an ideal response?
Answers may vary. Possible answer:
First, we show the statement is true when n = 1.
For n = 1, 81 > 1
So P1 is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is,
Pk: 8k > k is assumed true.
On the basis of the assumption that Pk is true, we need to show that Pk+1 is true.
Pk+1: 8k+1 > k + 1
So we assume that 
is true and multiply both sides of the equation by 8
8k? 8 > k8
8k+1 > k + 7k
8k+1 > k + 1 since 7k > 1
So Pk+1 is true if Pk is assumed true. Therefore, by the principle of mathematical induction, 
for all natural numbers n.
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when t = 0. Solve the differential equation of y when f(t) = 15t.
A. y = 750t + 37,500 + 47,500e-.02t B. y = -750t - 37,500 + 47,500e-.02t C. y = -750t - 37,500 + 47,500e.02t D. y = 750t - 37,500 + 47,500e-.02t
Find the common difference, d, for the arithmetic sequence.18, 14, 10, 6, . . .
A. 4 B. -6 C. -4 D. -12
Provide an appropriate response.When any nonzero number is divided by 1, the result is 
.
A. the nonzero number B. 0 C. 1 D. impossible to compute
A geometric sequence is given. Find the common ratio and write out the first four terms.{dn} = 
A. r = 
; d1 = 
, d2 = 
, d3 = 
, d4 = 
B. r = 5; d1 = 
, d2 = 
, d3 = 
, d4 = 
C. r = 
; d1 = 
, d2 = 
, d3 = 
, d4 = 
D. r = 5; d1 = 
, d2 = 
, d3 = 
, d4 = 