Problem Solving

I know this is a very simple problem , as per exponent rule says

a^b + a^c not equal to (a)^b+c

So in the attached file how you would get 2^9.

u have to visualize like that

2^n+2^n=2^n+1.
if u sum all given datas till 2^7, it equals to 2^8.
from given formula 2^8+2^8=2^9

Thanks!

another way of solving this.

S = 2 + 2 + 2^2 + 2^3 .......... + 2^8 -------- EQ1

Multiply 2 on both sides

2S = 2*2 + 2^2 + 2^3 + 2^4 ......... + 2^9 ----------- EQ2

EQ2 - EQ1

2S - S = 2*2 + 2^9 - (2+2) (all other terms will get canclled)
S = 2^9

And yet another way -

We see that there is a geometrical progression (GP) in the equation -
2 + GP

GP here is 2^1 + 2^2 + .... + 2^8

The sum of a GP is a0 * (r^n -1) / (r-1)
where a0 is the first term of this GP = 2^1
n = number of elements in the series = 8
r = ratio of any two consecutive numbers in the series (a1/a0 or a2/a1...) = 2

Thus sum = 2 * (2^8 - 1) / (2-1)
= 2^9 - 2

Thus, 2 + GP = 2 + 2^9 -2 = 2^9

Looks longish but if you remember the formula, this can be helpful in any GP problem.

Thanks.