Question

the sum of a b + 2 and 2 a b - 3​

Answer 1

Step-by-step explanation:

ab + 2 + 2ab - 3

3ab -1

Hope it helps

Answer 2

$\bold\pink{❥ᴀ᭄ɴsᴡᴇʀ:⤵}$

Lemma: [math]a^n+a^{n-1}b+\ldots+b^n=\frac{a^{n+1}-b^{n+1}}{a-b}[/math].

Proof: obvious if you cross-multiply and cancel.

Justification: Take the related identity [math]1+x+\ldots+x^n=\frac{x^{n+1}-1}{x-1}[/math], plug in [math]x=a/b[/math], and multiply both sides by [math]b^n[/math].

Now, since that’s the [math]k[/math]th term of our original sum, that can be written:

[math]\sum_{k=0}^n \frac{a^{k+1}-b^{k+1}}{a-b}=\frac{1}{a-b}\sum_{k=1}^{n+1} (a^k-b^k)=\frac{1}{a-b}\left(\frac{a(a^{n+1}-1)}{a-1}-\frac{b(b^{n+1}-1)}{b-1}\right)[/math]