Question

Help on Binomial question please!

Answer 1

Finding coefficient of\[ x ^{r} \] how do you convert \[3(1-3x)^{-1} \to 3(3^{r})\] and \[(1+\frac{x}{4})^{-1} \rightarrow (-1)^{r}(\frac{1}{4})^{r}\]

Answer 2

what should we do here?

Answer 3

do you know how to convert from 3[(1−3x)]^(-1) to 3(3r) ?

Answer 4

Its binomial expansion, Im suppose to find the coefficient

Answer 5

sorry

Answer 6

nvm its ok tq ^^

Answer 7

same to u
because i should not have open yur question when i dont about it :)

Answer 8

you may use binomial expansion for sure, but there is a nice trick here if we're allowed to use infinite geometric series :

\[\large 3(1-3x)^{-1} = \dfrac{3}{1-(3x)} = 3 + 3(3x) + 3(3x)^2 + \cdots \]

Answer 9

does that look familiar to you ? :)

Answer 10

is there a normal way of solving it ? ><

Answer 11

personally i feel infinite geometric series is the shortest+simplest way to solve this, haven't you studied geometric series yet ?

Answer 12

yes ive learn it

Answer 13

\[\large a+ar+ar^2+\cdots = \dfrac{a}{1-r}~~~\text{when |r|<1}\]

Answer 14

good :) then i suppose you're allowed to use it ^^

Answer 15

then how you get \[(-1)^{r}(\frac{1}{4})^{r} from (1+\frac{x}{4})^{-1}\]

Answer 16

use the same trick-
first write it as a fraction

Answer 17

\[\large \left(1+\frac{x}{4}\right)^{-1} = \dfrac{1}{1+\dfrac{x}{4}} = \dfrac{1}{1-\left(\color{red}{-\dfrac{x}{4}}\right)} \]

Answer 18

expand \(a=1, r = \color{red}{-\dfrac{x}{4}}\)

Answer 19

hmm but this wouldnt work for power of -2 and above. thank you ganesh!