We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.

+0

# Sequences & Series Problem

0
430
9
+96

What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is $$9$$?

May 31, 2017

### 9+0 Answers

#1
+100516
+1

I believe this is correct......

When  r   =  -1/2    the series sums to    9 / [ 1 -  (-1/2)]  =  9 / (3/2)  =  18/ 3    =   6

May 31, 2017
#3
+96
+1

How did you get $$r=-\frac{1}{2}$$?

benjamingu22  Jun 1, 2017
edited by benjamingu22  Jun 1, 2017
#2
+1

Sum = 9 / [1 -(-4/5)]

Sum = 9 /[1 + 4/5]

Sum = 9 / 1.8

Sum = 5

Jun 1, 2017
#4
+96
+1

How did you get $$r=-\frac{4}{5}$$?

benjamingu22  Jun 1, 2017
#5
0

They guessed at it as follows: divide the first term, 9 in this case, by one of the integers from 1 and up and see if it gives a ratio that converges. So, 9/6 =1.5 - 1 =0.5. Because of the formula for the sum of an infinite series, which is: S = F / [1 - r], they had to make it negative to converge to the smallest integer possible. Hence, 9/[1 - (-1/2)] =9/[1 + 1/2] =6. The same for 4/5 =9/[1 - (-4/5)] =9/[1 + 4/5] =5.

Theoretically, you could choose -8 and you would get: 9/[1 - (-8)] =9/[1 +8] =1, the smallest positive integer.

Jun 1, 2017
edited by Guest  Jun 1, 2017
#7
+96
+1

Thank you, Guest.

benjamingu22  Jun 1, 2017
#6
+100516
+1

Thanks, guest....your sum is certainly less than mine....!!!!

Jun 1, 2017
#8
+100516
+1

Here's the way to figure this.....

Let  S =  9 / [ 1 + r]

Since  l r l  < 1   , then  1 +  l r l  <  2

Look  at the graph of these , here :

https://www.desmos.com/calculator/b5eiddmoz7

Substituting "x" for "r", "S"  reaches a  positive integer minimum  of 5 inside the shaded area when  x  = .8   =  r

Thus

S  =  9 /  [ 1 + .8 ]  =  5

And....re-writing this in a slightly different manner, we have  that

S  =  9 / [ 1  - r ]

S  =  9  / [ 1  - (- .8)]   =   9 /  [ 1  -  (-4/5) ]

So.....  r =  -4 / 5

Jun 1, 2017
edited by CPhill  Jun 1, 2017
#9
+96
+1

Thank you, CPhill.

benjamingu22  Jun 1, 2017