a) The first 5 terms: 7, 4, 1, -2, -5
The sequence has a common difference. Hence, it is arithmetic sequence
b) The first 5 terms: 2, 5, 9, 17, 33
it is neither
c) The first 5 terms: 3, 30, 300, 3000, 30000
it is a geometric sequence
Step-by-step explanation:
a) a(1) = 7
a(n) = a(n-1) -3 for n is greater than or equal to 2
when n= 2
a(2) = a(2-1) - 3
a(2) = a(1) - 3 = 7 - 3
a(2) = 4
a(3) = a(3-1) - 3
a(3) = a(2) - 3 = 4 - 3
a(3) = 1
a(4) = a(4-1) - 3
a(4) = a(3) - 3 = 1 - 3
a(4) = -2
a(5) = a(5-1) - 3
a(5) = a(4) - 3 = -2 - 3
a(5) = -5
The first 5 terms: 7, 4, 1, -2, -5
The difference between the terms = 4-7 = 1-4 = -3
The sequence has a common difference. Hence, it is arithmetic sequence
b) b(n) = 2*b(n-1) - 1 for n is greater than or equal to 2
b(1) = 2
b(2) = 2×b(2-1) - 1
b(2) = 2×b(1) - 1 = 2(2)-1
b(2) = 5
b(3) = 2×b(3-1) - 1
b(3) = 2×b(2) - 1 = 2(5) - 1 = 10-1
b(3) = 9
b(4) = 2×b(4-1) - 1 = 2×b(3) - 1 = 2(9) - 1
b(4) = 17
b(5) = 2×b(5-1) - 1 = 2×b(4) - 1 = 2(17) - 1
b(5) = 33
The first 5 terms: 2, 5, 9, 17, 33
The difference between the terms = 5-2 = 3, 9-5 =4, 17 - 9 = 8
The difference are not the same. Hence, no common difference (not an arithmetic sequence)
we find the common ratio:
5/2 = 2.5, 9/5 = 1.8, 17/9 = 1.89
No common ratio (not a geometric sequence)
Hence, it is neither
c) c(n) = 10×c(n-1) for n is greater than or equal to 2
c(1) = 3
c(2) = 10×c(2-1) = 10×c(1) = 10 × 3
c(2) = 30
c(3) = 10 ×c(3-1) = 10×c(2) = 10 ×30
c(3) = 300
c(4) = 10 ×c(4-1) = 10×c(3) = 10 ×300
c(4) = 3000
c(5) = 10 ×c(5-1) = 10×c(4) = 10 ×3000
c(5) = 30000
The first 5 terms: 3, 30, 300, 3000, 30000
The difference between the terms = 300 -30 = 290, 3000-300= 2700
The difference are not the same. Hence, no common difference (not an arithmetic sequence)
We find the common ratio:
30/3 = 10, 300/30 = 10, 3000/300 = 10
The common ratio are the same.
Hence, it is a geometric sequence