Given a1 2 and ak1 ak 4 a Write the first 5 terms of the

Given a_1 = -2 and a_k+1 = a_k + 4. a) Write the first 5 terms of the sequence. b) What type of sequence is determined by the terms in (a)? c) What is formula for the nth term? d) Find the twenty-first term. e) Compute the sum of the first twenty-one terms. Given a_1 = 243 and a_k+1 = 1/3 a_k. a) Write the first 5 terms of the sequence. b) What type of sequence is determined by the terms in (a)? c) What is formula for the nth term? d) Find the eleventh term. e) Compute the sum of the first twenty-one terms. f) Compute the infinite sum 243 + 81 + 27 + 9 + ...

Solution

a1 = -2 ; ak+1 = ak +4

a) first 5 terms:

a2 = -2+4 = 2 ; a3 = 2 +4 =6 ; a4 = 6+4 = 10 ; a5 = 10+4 = 14

b) Arithmetic sequence : commmon difference , d = 4

c) formula for nth term : an = a1 + (n-1)d = -2 + (n -1)4 = 4n - 6

d) 21st term

a21 = a1 + (n -1)d = -2 + (21 -1)4 = -2 + 80 = 78

e) sum of first 21 terms : S21 = n(a21 +a1)/2 =21(-2 +78)/2 = 798