1 1 2 4 8 16 n 2 21 23 25 27 215 n I finally g

(1) 1 + 2 + 4 + 8 + 16 + ... + n

(2) 2^1 + 2^3 + 2^5 + 2^7 + 2^15 + ... + n

I finally got about (1) that it can be simplified to \"2n-1\".

However, I really don\'t know how to solve (2).

Solution

For the question (2) ,the series given is 2^1+2^3+2^7+2^15+.......

The power of this series are Mersenne Prime Numbers say 3,7,15,31,63,127,...

In general a Mersenne Prime is obtained by the formula Mn= 2^n-1

so for n=1 we get 1

for n=2 we get 3

for n=3 we get 7

for n=4 we get 15

for n=5 we get 31

and so on ....

now we come to the series 2^1+2^3+2^7+2^15+....... = 2^(2-1) +2^(2^2-1) +2^(2^3-1)+2^(2^4-1)+....

so the general tern of this series would be 2^(2^n-1) = General Term

hence the problem