exponents-Gprep

Question

What is the value of  2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8?

Got the correct ans by the long long long approach..Shorter way??

asterixmatrix · Answer

we can consider this as 2 + GP series (2+ 2^2----+ 2^8)
 then sum of n terms in GP series for this will be 2(2^8 -1)/(2-1) = 510

so we get 510 +2 = 512

lagomez · Answer

This is more about recognizing a pattern:

2 + 2 = 2^2

2^2 + 2^2 = 2^2(1 + 1) because we factor out the 2^2
2^2(2) = 2^3

2^3 + 2^3 = 2^3 (1+1) = 2^4

you can pretty much stop after that and recognize that you will eventually get 2^8 + 2^8 = 2^9

There usually tends to a be a pattern in these type of questions.

study · Answer

What is GP and what is the formula about?

Bunuel, can you please explain this one.

Thank you.

h2polo · Answer

GP is short for Geometric Progression. You can check out the wikipedia article for a detailed explaination:

https://en.wikipedia.org/wiki/Geometric_progression

But don't get caught up on trying to remember formulas. The key is to understand what you are looking at.

Just like lagomez said, it is about pattern recognition:

Here is the pattern I used:

2+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8
Factor out the two:
2*(1+1+2+2^2+2^3+2^4+2^5+2^6+2^7)=2*(2+2^1+2^2+2^3+2^4+2^5+2^6+2^7)
Keep factoring out the two...
2*2*(2+2+2^2+2^3+2^4+2^5+2^6)
2*2*2*(2+2+2^2+2^3+2^4+2^5)
2*2*2*2*(2+2+2^2+2^3+2^4)
2*2*2*2*2*(2+2+2^2+2^3)
2*2*2*2*2*2*(2+2+2^2)
2*2*2*2*2*2*2*(2+2)
2*2*2*2*2*2*2*2*(2)
OR
2^9=512

ctrlaltdel · Answer

study,

Here u go....hope it is useful.

Cheers

Kudos, if u feel it was helpful.

ctrlaltdel · Answer

One more...Source:tcy

Kudos if u find it helpful.

study · Answer

I tried to apply the GP formula but something is wrong - Can someone help please. Thanks.

a = the first term
r = the common ratio,
Tn = nth term and
Sn = sum of n terms

Sum of first n terms of G.P:
Sn =   / a(r - 1)
where r > 1

Sum 
  = 2 (2^10 - 1) = 2 * (1024 -1) = 2*1023
a(r - 1) = 2 ( 2 - 1) = 2

Hence sum = 2 * 1023 / 2 = 1023 that is incorrect

Bunuel · Answer

Sum of the terms of geometric progression is given by:  Sum=\frac{a*(r^{n}-1)}{r-1} , where  a  is the first term,  n  # of terms and  r  is a common ratio  >1 .

In our original question we have 2 plus G.P. with 8 terms, so:

2+(2+2^2+2^3+2^4+2^5+2^6+2^7+2^8)=2+\frac{2*(2^{8}-1)}{2-1}=1024

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