2.)The sum of a geometric series is -1023 and its first term is -3. If the last term is -768, what is the common ratio?
please supply you work if you have the time! i would like to understand this, as well as get a good grade!
You have a number of clues to help solve this..
Begin with the formula for the sum of a geometric series
$\displaystyle \displaystyle\ S_n=\frac{a\left(1-r^n\right)}{1-r}=\frac{a\left(r^n-1\right)}{r-1}$
Hence, you need three values.. $\displaystyle a,\;\;\;r,\;\;\;r^n$
You are given "a" and two clues to find the other two values.
$\displaystyle S_n=-1023$
$\displaystyle S_n=a+ar+ar^2+ar^3+......+ar^{n-1}$
hence you can find $\displaystyle r^{n-1}$ using the information given for the last term.
Also
$\displaystyle r^n=(r)r^{n-1}$
If you use these clues, you will get an equation in "r" which finds the value of "r".
Here is the Solution Hope this will help you,
Given a = -3
ar^{n} = -768
r^{n} = -768 / -3 = 256
so s = a (1 - r^{n}) / (1 - r)
change the sign on rhs
- 1023 = -3 (256 -1) / ( r -1)
r - 1 = -3(255)/ (-1023)
r = 1.7475 is the common ratio
If you need to know the value of n
1.7478....* n = 256
n log(10) (1.7478) = log (10) 256
n* 0.2425 = 2.4082
n = 9.93 approximately 10