Write the word or phrase that best completes each statement

Write the word or phrase that best completes each statement or answer the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. Does the following formula __________ 6 + 12 +18 +... + 6n - 3n(n +1) obeys Condition II of the Principle of Mathematical Induction, that is, if the formula is true for some natural number k, it is also true for the next natural number k = 1? Choose your answer appropriately. No, that is, the formula is true for some natural number k does not imply it is also true for the next natural number k + 1. Yes, that is, the formula is true for some natural number k implies that it is also true for the next natural number k + 1. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write out the first five terms of the sequence. {c_n} = {2n + 2/2n} c_1 = 3/2, c_2 = 4/3, c_3 = 5/4, c_4 = 6/5, c_5 = 7/6 c_1 = 3, c_2 = 4, c_3 = 5, c_4 = 6, c_5 = 7 c_1 = 2, c_2 = 3, c_3 = 4/3, c_4 = 5, c_5 = 6 c_1 = 2, c_2 = 3/2, c_3 = 4/3, c_4 = 5/4, c_5 = 6/5 write out the sum. sigma^n _k = 1(k + 7) 8 + 9 + 10 +...+ (n + 7) 1 + 2 + 3 +..+ n 7 + 8 + 9 +...+ (n + 7) n + 7 Solve Given that a_1 = -4, a_2 = -4 and a_n+2 = a_n - 1 - 4a_n, what is the fifth term of this recursively defined sequence? A_5 = 28 a_5 = -20 a_5 = 764 a_5 = -308 Find the sum. 1 + 3 + 5 + ... + 203 10, 201 10, 404 10, 609 10, 506 Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are a = 5; r = 4 pi a_5 = 1280 pi^4, a_n = 5 middot 4^n - 1 pi^n - 1 a_5 = 5 + 16 pi, a_n = 5 + 4 pi(n - 1) a_5 = 1280pi, a_n = 5 middot 4^n - 1 pi a_5 = 5120 pi^5, a_n = 5 middot 4^n pi^n

Solution

18) Here a1 = -4, a2 = -4

Then for n = 1, a3 = a2 - 4a1 = -4 + 16 = 12

Then a4 = a3 - 4a2 = 12 - 4*-4 = 28

Then a5 = a4 - 4a3 = 28 - (4*12) = 28 - 48 = -20

19) 1 + 3 + 5 + ........... 203 = 1 + 3 + 5 + ........... (102^2 -1)= 102^2 = 10404