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Transcript
How to sum a geometric series - The Easy Way.
Unlike how I made it seem during section, summing geometric series is very easy!
Sum of finite geometric series = S n = a + ax + ax 2 + ... + ax n −1 =
a (1 − x n )
1− x
(first term)[1- (ratio)(number of terms) ]
1- (ratio)
Therefore just find these three values and stick them into the above formula and that’s all!
IN PLAIN ENGLISH THIS MEANS: Sum =
So 5 + 5(2) + 5(2) 2 + 5(2)3 + 5(2) 4 + 5(2)5 + 5(2)6 =
n = number of terms = 7
a = first term = 5
therefore Sum =
x = ratio = 2
5(1 − 27 )
= 635
1− 2
What is 5(2)2 + 5(2)3 + 5(2) 4 = ?
It looks like cx 2 + cx 3 + cx 4 where c = 5 and x = 2
But don’t be fooled!! a = first term is not equal to c = 5
Instead a = first term = 5(2) 2 = 20
This one is just as easy as first one. Just use number of terms, first term, and ratio! It’s that easy!!
n = number of terms = 3
a = first term = 5(2) 2 = 20
x = ratio = 2
therefore Sum =
20(1 − 23 )
= 140
1− 2
(The reason this works is because the above series is really 20 + 20(2) + 20(2)2 in disguise. But you don’t ever need to figure
that out. Just use number of terms, first term, and ratio and stick those three numbers into the formula! You’ll get the correct
answer every time!! It’s that easy!!)
So on homework problem 9.2.26.b
What does Pn = 250(0.04) + 250(0.04) 2 + ... + 250(0.04) n −1 = ?
n = number of terms = n − 1
a = first term = 250(0.04)
x = ratio = 0.04
therefore Pn =
( 250(0.04) ) [1 − 0.04n −1 ]
1 − 0.04
=
(first term)[1- (ratio)(number of terms) ]
1- (ratio)
If we wanted to solve for Qn = 250 + 250(0.04) + 250(0.04) 2 + ... + 250(0.04)n −1 = ? then
n = number of terms = n
a = first term = 250
x = ratio = 0.04
therefore Qn =
( 250 ) [1 − 0.04n ]
1 − 0.04
=
(first term)[1- (ratio)(number of terms) ]
1- (ratio)
See how easy it is?
Summing an infinite series is also easy. For an infinite series, just find two numbers, (the first term and the ratio) and stick
a
(first term)
them into the following formula. Sum of infinite series =
=
.
1− x
1-(ratio)
But beware this only works when x = ratio < 1 . If x ≥ 1 ,the series can not be summed up. The sum will get bigger and
bigger and bigger or the sum will never settle down to one value.
Your TA,
Chris Deotte