Solve this problem that involves induction

[chwala]

Homework Statement

See attached.

Relevant Equations

Mathematical induction

Hello,
This is the attachment, the steps to solution are pretty clear. I guess there is a mistake on the highlighted part that prompts this thread.

Ought to be $3^{n+1} (n+2)-6$ and not $3^n(n+2)-6$. Unless i missed something, on another note, i find the first method (induction) better than second one (method of differences).

 

[pasmith]

Another method is to define $$f(x) = \sum_{k=1}^n x^k = \frac{x^{n+1}-x}{x-1}$$ so that $$xf'(x) = \sum_{k=1}^n kx^k$$ and we are looking to calculate $$\sum_{k=1}^n 3^k(2k+5) = 6f'(3) + 5f(3).$$

 

[Mark44]

I guess there is a mistake on the highlighted part that prompts this thread.

Ought to be $3^{n+1} (n+2)-6$ and not $3^n(n+2)-6$. Unless i missed something

I don't see any highlighted part.

 

[pasmith]

I don't see any highlighted part.

There is a faint red box around the $3^n$ in the last expression on the right. It should read $3^{n+1}$ as in the line above.

 

[Mark44]

There is a faint red box around the $3^n$ in the last expression on the right. It should read $3^{n+1}$ as in the line above.

I see it now after you pointed it out, but that's very faint.

 

[chwala]

Another method is to define $$f(x) = \sum_{k=1}^n x^k = \frac{x^{n+1}-x}{x-1}$$ so that $$xf'(x) = \sum_{k=1}^n kx^k$$ and we are looking to calculate $$\sum_{k=1}^n 3^k(2k+5) = 6f'(3) + 5f(3).$$

This is new to me, @pasmith. Can you continue with the next lines? Cheers, man.

 

[Gavran]

This is new to me, @pasmith. Can you continue with the next lines? Cheers, man.

Express $f(3)$ and $f’(3)$ in terms of $n$ by using $$ f(x)=\frac{x^{n+1}-x}{x-1} $$ and $$ f'(x)=\frac{d}{dx}(\frac{x^{n+1}-x}{x-1}) $$.

 

[chwala]

I don't see any highlighted part.

@Mark i re-uploaded, i too could not see the highlighted part