Question

Expand (1+x)^10 up to and including the term in x^3.Hence,obtain an approximation for (1.01)^10 and (0.99)^10

Answer 1

Hw shld I begin

Answer 2

write out the binomial expansion formula

Answer 3

\[\large (x+1)^{10} = \sum \limits_{k=0}^{10} \binom{10}{k}x^k1^{10-k} \]

\[\large = \sum \limits_{k=0}^{10} \binom{10}{k}x^k \]

Answer 4

expand the sum ^^

Answer 5

hw to expand

Answer 6

know how to expand below :

\[\large \sum \limits_{k=0}^3 k\]

?

Answer 7

nope

Answer 8

its easy :

\[\large \sum \limits_{k=0}^3 k \]

is just a lazy way of writing :

\[\large 0+1+2+3\]

Answer 9

\[\large \sum \limits_{k=0}^3 k = 0+1+2+3 \]

Answer 10

based on that, can you guess what below expansion would be :

\[\large \sum \limits_{k=0}^3 k^2 = ? \]

?

Answer 11

0+1+2+3

Answer 12

think again, you have \(\large k^2\) right ?

Answer 13

0+1+4+9

Answer 14

like this

Answer 15

exactly !


\[\large \sum \limits_{k=0}^3 k^2 = 0^2 + 1^2 + 2^2 + 3^2 \]

Answer 16

think of left hand side as a one full BOX full of gifts,
when u open the BOX, you will see so many gifts(terms on right hand side)

Answer 17

Alright, lets get back to the original problem :

\[\large \sum \limits_{k=0}^{10} \binom{10}{k}x^k\]

\[\large = \binom{10}{0}x^0 +\binom{10}{1}x^1 +\binom{10}{2}x^2 +\binom{10}{3}x^3 + \cdots \]

Answer 18

see if above expansion makes sense..
just find the coefficients and plug in ^^

Answer 19

yes..

Answer 20

1

Answer 21

what/who/when 1 ?

Answer 22

find the remaining coefficients also and plug in