SOLUTION: The third term of a geometric progression is nine times the first term.The sum of the first four terms is K times the first term. Find the values of possible values of K.

Click here to see ALL problems on Sequences-and-series

Question 1151377: The third term of a geometric progression is nine times the first term.The sum of the first four terms is K times the first term. Find the values of possible values of K.
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Let's say that 'a' is the first term. To get the next term, we multiply by r. We keep this pattern up to generate as many terms as we want.

first term = a
second term = (first term)*r = a*r
third term = (second term)*r = (a*r)*r = ar^2
fourth term = (third term)*r = (ar^2)*r = ar^3
and so on...

Now use the fact that the third term is 9 times the first term
third term = 9*(first term)
third term = 9*a
ar^2 = 9*a
r^2 = 9 .... divide both sides by 'a'
r = 3 .... apply square root to both sides

Use this value of r to compute the first four terms
first term = a
second term = a*r = a*3 = 3a
third term = ar^2 = a(3)^2 = 9a
fourth term = ar^3 = a(3)^3 = 27a

Now add them up
sum of 1st four terms = (1st term)+(2nd term)+(3rd term)+(4th term)
sum of 1st four terms = (a)+(3a)+(9a)+(27a)
sum of 1st four terms = 40*a

We're told that the sum of the first four terms is equal to k times the first term, so set 40*a equal to k*a, and divide both sides by 'a'
k*a = 40*a
k = 40

Answer: k = 40

Answer by ikleyn(52833) (Show Source):
You can put this solution on YOUR website!
.

We are given = = . It implies = 9; hence, taking the square root of both sides, we have TWO possible values for the common ratio "r" : r = 3 or r = -3. If r= 3, then the sum of the first four terms is = = . Hence, in this case K = 40. If r= -3, then the sum of the first four terms is = = . Hence, in this case K = -20. ANSWER. Two values are possible for K: 40 and - 20.

Solved.