given geometric series is 256+p+64-32 determine the value of p?
p = 128
To determine the value of p in the given geometric series, we need to understand the pattern and properties of a geometric series.
A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant factor called the common ratio (r).
The general formula for the sum of a geometric series with the first term (a) and the common ratio (r) is:
Sn = a * (1 - r^n) / (1 - r)
Where:
- Sn is the sum of the first n terms of the series
- a is the first term of the series
- r is the common ratio
- n is the number of terms in the series
In the given series: 256 + p + 64 - 32, we can identify that the first term (a) is 256 and the common ratio (r) is -1/2.
Now, let's determine the number of terms in the series (n).
n = number of terms = 4 (as there are four terms in the series: 256, p, 64, -32)
Using the formula for the sum of a geometric series, we can find the sum of the series (Sn):
Sn = a * (1 - r^n) / (1 - r)
= 256 * (1 - (-1/2)^4) / (1 - (-1/2))
= 256 * (1 - 1/16) / (3/2)
= 256 * (15/16) / (3/2)
= 256 * (15/16) * (2/3)
= 320
We know that the sum of the series (Sn) is equal to 320. Now, we subtract the known terms in the series from the sum to find the value of p:
320 = 256 + p + 64 - 32
Rearranging the equation to isolate p:
p = 320 - 256 - 64 + 32
= 32
Therefore, the value of p in the given series is 32.