I'm sort of there but getting a bit confused!! This is the question:
a) Classify the following sequences as geometric, arithmetic or neither and justify your choice.
i) 1,1,2,3,5, ...
ii) 1.2, 1.1, 1, 0.9, 0.8, ...
iii) 0, 1, 2, 5, 26, ...
iv) 3,6, 12, 24, 48, ...
v) 10, 8, 6.4, 5.12, ...
b) Which of the above geometric sequences has a sum to infinity? Find this sum.
c) Find a formula for the sum of the first n terms of the geometric sequence in part (b) and show how this relates to the sum of infinity.
Any help greatly appreciated!!
Thank you.
a) Classify the following sequences as geometric, arithmetic or neither and justify your choice.
i) 1,1,2,3,5, ...
ii) 1.2, 1.1, 1, 0.9, 0.8, ...
iii) 0, 1, 2, 5, 26, ...
iv) 3,6, 12, 24, 48, ...
v) 10, 8, 6.4, 5.12, ...
b) Which of the above geometric sequences has a sum to infinity? Find this sum.
c) Find a formula for the sum of the first n terms of the geometric sequence in part (b) and show how this relates to the sum of infinity.
Any help greatly appreciated!!
Thank you.
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ai) This is a Fibonnaci series; each term is the sum of the previous two. It is "neither."
aii) This is an arithmetic series; the difference between terms is -0.1. It is "arithmetic."
aiii) This series is formed by n^2 + 1, where n is starts with 0 and each term is computed from the previous using the formula. It is "neither."
aiv) This is a geometric series; each term is derived by multiplying the previous by 2. It is "geometric."
av) This is a geometric series; each term is derived by multiplying the previous by 0.8. It is "geometric."
b) Only the last has a sum to infinity, since the absolute value of its multiplier is less than 1.
c) Formula for sum of geometric series: a(1-r^n) / (1-r), where a is the first term, r is the multiplier and n is the number of terms. If you let n go to infinity, the formula reduces to a / (1-r) which would be 10 / (1-0.8) = 10 / 0.2 = 50 for the last example above.
aii) This is an arithmetic series; the difference between terms is -0.1. It is "arithmetic."
aiii) This series is formed by n^2 + 1, where n is starts with 0 and each term is computed from the previous using the formula. It is "neither."
aiv) This is a geometric series; each term is derived by multiplying the previous by 2. It is "geometric."
av) This is a geometric series; each term is derived by multiplying the previous by 0.8. It is "geometric."
b) Only the last has a sum to infinity, since the absolute value of its multiplier is less than 1.
c) Formula for sum of geometric series: a(1-r^n) / (1-r), where a is the first term, r is the multiplier and n is the number of terms. If you let n go to infinity, the formula reduces to a / (1-r) which would be 10 / (1-0.8) = 10 / 0.2 = 50 for the last example above.