Calculate the sum of the following series: -6 + 24 – 96 + … + 98304
It is clearly a GP, with
a = -6
r = -4
98304/6 = 16384 = 4^7
so there are 8 terms, and
S8 = a(r^8-1)/(r-1) = -6((-4)^8 - 1)/(-4-1) = 78642
Please help, I don't understand how to do this
To calculate the sum of the series -6 + 24 - 96 + ... + 98304, we notice that each term is obtained by multiplying the previous term by -4.
First, let's find the common ratio of the series:
Common ratio = (-24) / (-6) = 4
Next, let's find the number of terms in the series. We can do this by finding the power to which the common ratio is raised to get the last term:
Last term = 98304 = (-6) * (-4)^n
Divide both sides by -6:
(-4)^n = -98304 / -6
(-4)^n = 16384
To find the value of n, we take the logarithm of both sides:
n * log(-4) = log(16384)
n = log(16384) / log(-4)
Since the logarithm of a negative number is undefined, we conclude that there is no whole number solution for n. Thus, we must consider the partial sum.
Now, let's find the partial sum S_n using the formula for the sum of a geometric series:
S_n = a * (1 - r^n) / (1 - r)
In this case, a = -6 and r = -4. Let's plug these values into the formula:
S_n = -6 * (1 - (-4)^n) / (1 - (-4))
Since n is not a whole number, we cannot calculate the exact sum of the series. However, we can evaluate partial sums for specific values of n.
To calculate the sum of the given series, we need to find the pattern in the terms and then apply the formula for the sum of a geometric series.
In this series, each term is obtained by multiplying the previous term by -4. So, the pattern is:
-6, -6 * -4 = 24, 24 * -4 = -96, -96 * -4 = 384, 384 * -4 = -1536, ...
We can see that the common ratio is -4, and the first term is -6. The last term given is 98304.
To find the sum of a geometric series, we use the formula:
Sum = (First Term * (1 - r^n)) / (1 - r)
Where:
- First Term is the first term of the series (-6 in our case).
- r is the common ratio (-4 in our case).
- n is the number of terms.
We need to find n, the number of terms. Using the formula to calculate each term recursively, we start with the first term and keep multiplying by -4 until we get to 98304. This process can be done by dividing the term by -4 repeatedly until we reach 1:
98304 ÷ -4 = -24576
-24576 ÷ -4 = 6144
6144 ÷ -4 = -1536
-1536 ÷ -4 = 384
384 ÷ -4 = -96
-96 ÷ -4 = 24
24 ÷ -4 = -6
From this, we can determine that there are 7 terms in the sequence, including -6 and 98304.
Now, we substitute the values into the formula:
Sum = (-6 * (1 - (-4)^7)) / (1 - (-4))
Using the exponent calculation:
Sum = (-6 * (1 - 16384)) / (1 + 4)
Simplifying further:
Sum = (-6 * (-16383)) / 5
Sum = 6 * 16383 / 5
Sum = 98098 / 5
Sum = 19619.6
Therefore, the sum of the series -6 + 24 - 96 + ... + 98304 is 19619.6.