Arithmetic In Arithmetic And Power Of Number

Question

How do i solve arithmetic problem that contain another arithmetic in it and the arithmetic is problem is using power of number?

This is the problem that i run into $(2^1) + (2^1 + 2^2) + (2^1 + 2^2 + 2^3) ... U_n$

Is it possible to find the Un and Sn ?

Answer 1

Hint:

$1=2^1-1$

$1+2 = 2^2-1$

$1+2+4 = 2^3-1$

$1+2+4+8 = 2^4-1$

$1+2+4+8+16 = 2^5-1$

$1+2+4+\dots+2^n = 2^{n+1}-1$

So then, what is $U_n$?

Now, looking at $S_n$, can you see a similar pattern emerging?

---

Solution:

Given that $1+2+4+\dots+2^n = 2^{n+1}-1$ it follows that $U_n = 2+4+\dots+2^n = 2^{n+1}-2$

Further, we have that $S_n = U_1+U_2+\dots+U_n$

By replacing $U_n$ with what we found earlier and rearranging we have:

$S_n = (2^2-2)+(2^3-2)+\dots+(2^{n+1}-2) = (2^2+2^3+\dots+2^{n+1})-2n$

Recalling that $1+2+4+\dots+2^n+2^{n+1}=2^{n+2}-1$ and by subtracting $1+2$ from each side, we arrive at:

$S_n = (2^{n+2}-4)-2n$