To evaluate the summation of the expression 20 * 0.5^(n – 1) from n = 3 to n = 12, we start by setting up the summation:
∑ (from n=3 to n=12) 20 * 0.5^(n – 1)
This means we will calculate the value for each integer n from 3 to 12 and then sum those values together.
First, let’s break it down:
- When n = 3: 20 * 0.5^(3 – 1) = 20 * 0.5^2 = 20 * 0.25 = 5
- When n = 4: 20 * 0.5^(4 – 1) = 20 * 0.5^3 = 20 * 0.125 = 2.5
- When n = 5: 20 * 0.5^(5 – 1) = 20 * 0.5^4 = 20 * 0.0625 = 1.25
- When n = 6: 20 * 0.5^(6 – 1) = 20 * 0.5^5 = 20 * 0.03125 = 0.625
- When n = 7: 20 * 0.5^(7 – 1) = 20 * 0.5^6 = 20 * 0.015625 = 0.3125
- When n = 8: 20 * 0.5^(8 – 1) = 20 * 0.5^7 = 20 * 0.0078125 = 0.15625
- When n = 9: 20 * 0.5^(9 – 1) = 20 * 0.5^8 = 20 * 0.00390625 = 0.078125
- When n = 10: 20 * 0.5^(10 – 1) = 20 * 0.5^9 = 20 * 0.001953125 = 0.0390625
- When n = 11: 20 * 0.5^(11 – 1) = 20 * 0.5^{10} = 20 * 0.0009765625 = 0.01953125
- When n = 12: 20 * 0.5^(12 – 1) = 20 * 0.5^{11} = 20 * 0.00048828125 = 0.009765625
Now, we sum all these values:
Total = 5 + 2.5 + 1.25 + 0.625 + 0.3125 + 0.15625 + 0.078125 + 0.0390625 + 0.01953125 + 0.009765625
Calculating this gives:
Total ≈ 9.09375
Therefore, the evaluated summation of 20 * 0.5^(n – 1) from n = 3 to n = 12 is approximately 9.09375.