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Binomial expansion help please

Given that the expansion (1+ax)^n begins with 1+36x+576x^2 find the values of a and n

Original post by osayuki

Given that the expansion (1+ax)^n begins with 1+36x+576x^2 find the values of a and n

Can you expand

(1+ax)^n

in terms of

a

and

n

?

It starts

(1+ax)^n = 1 + nax + ...

Then compare the coefficients.

E.g. the second term in the expansion is

36x

so

na = 36

.

Does that make sense?

OP

Original post by notnek

Can you expand

(1+ax)^n

in terms of

a

and

n

?

It starts

(1+ax)^n = 1 + nax + ...

Then compare the coefficients.

E.g. the second term in the expansion is

36x

so

na = 36

.

Does that make sense?

Thank you for replying. I'm unsure how to get nax as I couldn't figure out how to cancel n!/(n-1!)1!

Original post by osayuki

Thank you for replying. I'm unsure how to get nax as I couldn't figure out how to cancel n!/(n-1!)1!

\frac{n!}{(n-1)!}

(We can "ignore" the 1! since it just equals 1)

Remembering what factorial actually means and writing n! and (n-1)! out fully we get;

\frac{n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1}{(n-1) \times (n-2) \times ... \times 3 \times 2 \times 1}

Cancelling terms which occur on both lines are left with;

n

OP

Original post by DylanJ42

\frac{n!}{(n-1)!}

(We can "ignore" the 1! since it just equals 1)

Remembering what factorial actually means and writing n! and (n-1)! out fully we get;

\frac{n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1}{(n-1) \times (n-2) \times ... \times 3 \times 2 \times 1}

Cancelling terms which occur on both lines are left with;

n

Thank you so much :smile:

:smile:

Original post by osayuki

Thank you for replying. I'm unsure how to get nax as I couldn't figure out how to cancel n!/(n-1!)1!

It will make your life easier if you remember the pattern of the coefficients so you don't have to derive each one.

E.g. for the simple expansion (1+x)^n, the second term coefficient is 'n', the third is n(n-1)/2!, the fourth is n(n-1)(n-2)/3! etc.

This may be on your formula book.

[QUOTE=notnek;62561237]It will make your life easier if you remember the pattern of the coefficients so you don't have to derive each one.

E.g. for the simple expansion (1+x)^n, the second term coefficient is 'n', the third is n(n-1)/2!, the fourth is n(n-1)(n-2)/3! etc.

This may be on your formula book.
Thank you again for your help

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