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# Binomial Theorem

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How to solve this

For some real number a and some positive integer n, the first few terms in the expansion of (1 + ax)^n are

1 + 10x + 150 x^2 + cx^3 + ...

Find c.

Feb 11, 2023

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We can use the binomial theorem to find the coefficient of the term \(x^3\) in the expansion of \( (1 + ax)^n\).

The binomial theorem states that, for any real number a and any positive integer n, the expansion of (1 + ax)^n is given by:

\((1 + ax)^n = C(n, 0) * 1^(n-0) * (ax)^0 + C(n, 1) * 1^(n-1) * (ax)^1 + C(n, 2) * 1^(n-2) * (ax)^2 + ... + C(n, n) * 1^(n-n) * (ax)^n\)

where C(n, k) is the binomial coefficient given by:

\(C(n, k) = n! / (k! (n-k)!)\)

where n! means n factorial, \(i.e., n! = n * (n-1) * (n-2) * ... * 1\).

So, to find the coefficient of x^3 in the expansion of \((1 + ax)^n\), we need to find \(C(n, 3) * 1^(n-3) * (ax)^3\).

Since \( 1^(n-3) = 1\), the coefficient of \(x^3\) is simply \(C(n, 3) * ax^3 = n! / (3! (n-3)!) * a^3\).

Therefore, if c is the coefficient of \(x^3\) in the expansion of \((1 + ax)^n\), we have:

\(c = n! / (3! (n-3)!) * a^3\)

So, to find c, we need the values of n and a. Unfortunately, the problem doesn't give us these values, so we cannot find c without more information.

Feb 11, 2023