What is the first term in a geometric series with ten terms, a common ratio of 0.5 and a sum of 511.5?
I got 256 as my answer.
511.5=a1[1-(.5)^10]/1-1/2
511.5=a1(1023/1024)/1/2
511.5=a1(1023/512) ----> divide by both sides
256=a1
CAN SOMEONE PLZ TELLME THE ANSWER TO THIS QUESTION Complete the following stepped-out solution to show that 511⋅w⋅115+3 is equivalent to w+3.(1 point)
511⋅w⋅115+3
511⋅
⋅w+3 Commutative Property of Multiplication
⋅w+3 Inverse Property of Multiplication
w+3 Identity Property of Multiplication
The first term of a geometric series is 7 and the ratio between terms is 0.5. Find S4.
To find the first term (a1) of a geometric series with a common ratio of 0.5, we can use the formula for the sum of a geometric series:
S = a1 * (1 - r^n) / (1 - r)
Where:
- S is the sum of the series (511.5)
- a1 is the first term
- r is the common ratio (0.5)
- n is the number of terms (10)
Plugging in the given values, we have:
511.5 = a1 * (1 - 0.5^10) / (1 - 0.5)
Simplifying the equation:
511.5 = a1 * (1 - 0.0009765625) / 0.5
511.5 = a1 * (0.9990234375) / 0.5
Multiplying both sides by 0.5 to isolate a1:
255.75 = a1 * 0.9990234375
Dividing both sides by 0.9990234375:
255.75 / 0.9990234375 = a1
The first term, a1, is approximately 256.72, rounded to two decimal places.
To find the first term (a1) in a geometric series, you can use the formula for the sum of a geometric series:
Sn = a1(1 - r^n) / (1 - r)
In this case, we are given the values of Sn (the sum), n (the number of terms), and r (the common ratio). We can plug these values into the formula and solve for a1.
Given:
Sn = 511.5
n = 10
r = 0.5
511.5 = a1(1 - 0.5^10) / (1 - 0.5)
Let's simplify the expression inside the parentheses:
1 - 0.5^10 = 1 - 0.0009765625 = 0.9990234375
Substituting this value back into the equation:
511.5 = a1(0.9990234375) / (1 - 0.5)
Let's simplify the expression in the denominator:
1 - 0.5 = 0.5
Substituting this value back into the equation:
511.5 = a1(0.9990234375) / 0.5
Now, let's isolate a1 by multiplying both sides of the equation by 0.5:
0.5 * 511.5 = a1 * (0.9990234375)
255.75 = a1 * (0.9990234375)
Finally, divide both sides of the equation by (0.9990234375) to solve for a1:
255.75 / (0.9990234375) = a1
256 = a1
Therefore, the first term in the geometric series is 256.
Looks good to me.
You can also check by adding
256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 + 0.5