Answer:
- infinity (does not converge)
- infinity (does not converge)
- infinity (does not converge)
- 47 13/16
- -1023
Explanation:
You want the sums of various geometric series.
Sum of a geometric sequence
The sum of n terms of a geometric sequence with first term a1 and common ratio r is ...
Sn = a1·(r^n -1)/(r -1)
When the series is infinite, the sum will converge if and only if |r| < 1.
The ratio can be found as the ratio of the first two terms:
r = a2/a1
1. 8, 16, ...
The ratio is ...
r = 16/2 = 2
The magnitude of r is greater than 1, so this series does not converge.
2. 5, 25, ...
The ratio is ...
r = 25/5 = 5
The magnitude of r is greater than 1, so this series does not converge.
3. 1, 4, ...
The ratio is ...
r = 4/1 = 4
The magnitude of r is greater than 1, so this series does not converge.
4. 24, 12, ... S8
The ratio is ...
r = 12/24 = 1/2
The sum of the first 8 terms is ...
S8 = 24·((1/2)^8 -1)/(1/2 -1) = 24·(-255/256)/(-1/2) = 24(255/128)
S8 = 47 13/16
5. 3, -6, ... S10
The ratio is ...
r = -6/3 = -2
The sum of the first 10 terms is ...
S10 = 3·((-2)^10 -1)/(-2 -1) = 3(1023)/(-3)
S10 = -1023
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Additional comment
If you have a number of these, a spreadsheet or graphing calculator can do the math for you.