What is the unit digit of A = 2 + 2^2 + 2^3 + ... + 2^20 ?

Question

What is the unit digit of  A = 2 + 2^2 + 2^3 + ... + 2^{20}  ?

A. 0
B. 2
C. 4
D. 6
E. 8

chetan2u · Answer

Cyclicity of units digit of power of 2: 
 2^1=2,2^2=4,2^3=8,2^4=16,2^5=32,2^6=64 
So, the pattern of units digit is 2,4,8,6,2,4
Thus,  2^{4k}, 2^{4k+1}, 2^{4k+2},2^{4k+3}  have units digit of 2, 4, 8, and 6

How do we use in this question:

The powers 1 to 20 are 5 sets of (4K, 4K+1, 4K+2 and 4K+3):  (2^1,2^2,2^3,2^4);(2^5..2^8);(2^9…2^{12});(2^{13}….2^{16});(2^{17}…2^{20}) 
The units digit of  2^{4k}=6 \ \ 2^{4k+1}=2 \ \ 2^{4k+2}=4 \ \ 2^{4k+3}=8

Units digit of the entire sum = 5*(6+2+4+8) = 5*20 or 0.

A

PyjamaScientist · Answer

I solved the question by sum of a GP.
Can someone what Chetan sir has done here?

CrackverbalGMAT · Answer

Consider the power cycle of 2:

2^1 = 2  
 2^2 = 4  
 2^3 = 8  
 2^4 = 16 
 2^5 = 32 
 2^6 = 64 
You can see that the unit digit gets repeated after every fourth power of 2. Hence, you can say that 2 has a power cycle of 2, 4, 8, 6 with cyclicity 4.

What is the unit digit of A =  2 + 2^2 + 2^3 + ... + 2^20

The unit digit of the first 4 term in the series( 2^1,2^2,2^3,2^4 ) will be 2, 4 ,8 and 6 . Hence the unit digit of their sum would be 2+4+8+6 is 0.
The next 4 terms(  2^5, 2^6,2^7 ,2^8 ) will have the same unit digits as  2^1,2^2,2^3,2^4  due to cyclicity property i.e 2, 4 ,8, 6 and the unit digit of their sum will also be 0.

so, the 20 terms in the series can be divided into 5 groups with 4 terms in each group . Each group has the terms with unit digits 2,4,8 and 6. 
The unit digit of the sum of each group is 0. Therefore unit digit of A will also be 0.

Option A is the answer.

Thanks,
  Clifin J Francis,
GMAT SME

chetan2u · Answer

I have added the details. Please see if it helps you.

Berdiyor · Answer

2 + 4 + 8 + 16 = 30
32 + 64 + 128 + 256 = ...0
...

20 is divisible by 4. Therefore, the last digit ends with 0.

bumpbot · Answer

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What is the unit digit of A = 2 + 2^2 + 2^3 + ... + 2^20 ?

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What is the unit digit of A = 2 + 2^2 + 2^3 + ... + 2^20 ?

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