is there a mathematical equation for 2 + 2^2 + 2^3 + 2^4 + 2^5 ...

Question

Is there a mathematical equation for:

2 + 2^2 + 2^3 + 2^4 + 2^5 + ... 2^n

I'm not sure what this type of pattern is called.

This is called as geometric progression. https://en.wikipedia.org/wiki/Geometric_progression — Shraddheya Shendre, Nov 16 '16 at 19:31
Try calculating the result for $n=2,$ $n=3,$ $n=4,$ $n=5,$ etc. and see if you notice a pattern. — Dave L. Renfro, Nov 16 '16 at 19:33

score 3 · Answer 1 · answered Nov 16 '16 at 19:34

This is called a geometric series. There is a well-known formula for summing finite geometric series:

$$ar+ar^2+...+ar^n = \frac{ar^{n+1}-ar}{r-1}$$

Here's a proof:

Let $S = ar+ar^2+...+ar^n$. Then, we have that

$$rS = ar^2+ar^3+...+ar^{n+1}$$

Now, consider $rS-S$. Note that almost all of the terms cancel:

$$rS-S = (ar^{n+1}+ar^n+...+ar^2)-(ar^n+...+ar^2+ar) = ar^{n+1}-ar$$

So, we have that

$$S = \frac{ar^{n+1}-ar}{r-1}$$

Thus, in your case when $a=1,r=2$, we have the closed form

$$\frac{2^{n+1}-2}{1} = 2^{n+1}-2$$

score 1 · Accepted Answer · answered Nov 16 '16 at 19:34

1

It is the sum of the $n$ first terms of a geometric progression of ratio $2$ and first term $2$ and so one has

$$2+2^2+\cdots+2^n=2\times {2^n-1\over 2-1}=2^{n+1}-2$$

answered Nov 16 '16 at 19:34

marwalix

1

Considering the knowledge of the OP, this answer seems too jumpy. – Shraddheya Shendre Nov 16 '16 at 19:37
You're right, on my smartphone I edit answers in steps. I was giving a proof of the geometric progression formula when you made your comment – marwalix Nov 16 '16 at 19:40

score 1 · Answer 3 · answered Nov 16 '16 at 19:39

1

Let $S_n=2+2^2+2^3+...2^n$.

then

$$S_{n+1}=2+2^2+2^3+...2^n+2^{n+1}$$

$$=2(1+S_n)=S_n+2^{n+1}$$

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

answered Nov 16 '16 at 19:39

hamam_Abdallah

3 Answers3