Is this correct? (Last step $\rightarrow$ After taking L.C.M.)

$\large \dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + ... + \dfrac{b_n}{a_n} = \sum\limits_{k=0}^{n} \Big(\dfrac{b_k}{a_k}\Big) = \dfrac{\large{\sum\limits_{p=0}^{n}} \Big(\dfrac{b_p}{a_p}{\times \prod\limits_{q=0}^{n} a_q}\Big)}{\prod\limits_{q=0}^{n} a_q}$


And also feel free to add any extra thing to your answer!


This is true simply by linearity of summation: if $k$ is constant with respect to $p$ then $$\sum_{p=0}^n ka_p = k \cdot \sum_{p=0}^n a_p$$ Here, $k = \displaystyle \prod_{q=0}^n a_q$, which is constant with respect to $p$, so you can factor it out of the sum and cancel it.

You mention LCMs, though it doesn't look like you used them anywhere. In any case, your equation remains true if you replace $\displaystyle \prod_{q=0}^n a_q$ by $\mathrm{lcm} \{ a_q \mid 1 \le q \le n \}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.