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Hello,

I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise.

What I need to show is the following:

$$

(a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j}

$$

My attempt, starting from the LHS. Expanding the brackets:

$$

(a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = a \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} + b \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j}

\\ = \sum_{j = 0}^{n} \binom nj a^{n+1-j}b^{j} + \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j+1}

$$

Now I know a useful equality (that I proved earlier) is:

$$

\binom{n+1}{k} = \binom{n}{k} + \binom{n}{k-1}

$$

However I am not sure how to proceed. Any tips would be much appreciated!

I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise.

What I need to show is the following:

$$

(a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j}

$$

My attempt, starting from the LHS. Expanding the brackets:

$$

(a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = a \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} + b \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j}

\\ = \sum_{j = 0}^{n} \binom nj a^{n+1-j}b^{j} + \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j+1}

$$

Now I know a useful equality (that I proved earlier) is:

$$

\binom{n+1}{k} = \binom{n}{k} + \binom{n}{k-1}

$$

However I am not sure how to proceed. Any tips would be much appreciated!

Last edited: