Question

Question..
how much does the 2 formulas for finding geometric series(G.P) matters?

Answer 1

The 2 formulas are \[\frac{a(1-r ^{n})}{1-r}, r<1 \] and \[\frac{a(r ^{n}-1)}{r-1}, r>1\]

Answer 2

I have used the 1st formula even when r > 1 and it still works. Can someone explain the concept to me ? ><

Answer 3

First formula can also work as you are just multiplying and dividing by (-1) there.. :)

Answer 4

\[\frac{a(r^n - 1)}{r - 1} \times \frac{-1}{-1} \implies \frac{a(1 - r^n)}{1 - r}\]

Answer 5

but why does the textbook says one is for r > 1 and the other is for r < 1 ? =.=

Answer 6

Okay..

Answer 7

Suppose your r is negative, means r < 1..

Answer 8

And n = 2..

Answer 9

Then you should use first formula here :

\[\frac{a(1 - (-2)^{2})}{1 - (-2)} \implies \frac{-3a}{3} \implies -a\]

Answer 10

Now let us use second formula here:

\[\frac{a[(-2)^2 - 1]}{-2 -1} \implies -a\]

Answer 11

And I am gone.. :)

Answer 12

It has become my doubt too now.. :P

Answer 13

Don't worry, Ganesh will take us to the right path. :)

Answer 14

These are partial sum formulas, so they don't depend on wheter \(r\) is less than 1 or not. it seems the textbook is trying to confuse you by oversimplifying formulas haha! just memorize any one of them as both of them are equivalent as water has shown in the first reply :

\[\large \frac{a(r^n - 1)}{r - 1} =\frac{a(1 - r^n)}{1 - r}\]

Answer 15

oh i see, thank you! :)

Answer 16

r forever cant be 1 right ?

Answer 17

we don't want r to be 1, yes :) you knw why !

Answer 18

okay nvm.. thats a stupid question =.=

Answer 19

is it possible for r to be negative ?

Answer 20

@ganeshie8 I have also read in the textbook stating r<1 use this and for r>1 use that like but I have also observed that the both the formulas can be used..

It is just like Slope formula, where you can subtract y_2 from y_1 also..

Answer 21

because the text book says mod r < 1

Answer 22

2, -4, 8, -16

Find here the common ratio.. :)

Answer 23

um.. ok tq ><

Answer 24

thats a good thing to compare(slope formula)
you must be looking at `converging infinite series` formula :
\[\large a + ar + ar^2 + \cdots = \dfrac{a}{1-r}~~~\color{red}{\text{when |r| < 1}}\]

Answer 25

As far as geometric series/sequence is concerned, there are only 3 formulas to comprehend+memorize :

1) nth term of sequence : \(\large a_n = ar^{n-1}\)

2) partial sum of series : \(\large \dfrac{a(1-r^n)}{1-r}\)

3) infinite sum of a converging series : \(\large \dfrac{a}{1-r}~~~\color{red}{\text{when |r| < 1}}\)