What is the units digit of the sum 1+5+5^2+5^3+5^4+...+5^2012+5^2013?

Question

I am not sure how to find the answer to this. A hint said: first, find the sum 1+5 then the sum 1+5+5^2, then the sum 1+5+5^2+5^3. Do you observe a pattern? Describe what happens and why. 

Answer 1

1 + 5 + 5² + 5³ + .... + 5²⁰¹³

 

This is the sum of powers of 5

Note that 5⁰ = 1

 

As the hint said, look at the first few terms:

1

1+5 = 6

6 + 5² = 6 + 25 = 31

31 + 5³ = 31 + 125 = 156

156 + 5⁴ = 156 + 625 = 781

 

Your pattern on the ones digit is 1, 6, 1, 6, 1, ....

You actually have 2014 terms in the summation with going from n = 0 to n = 2013

The 2014th term will end in 6

 

The reason is that each term after the initial 1 will be adding a power of 5.

All powers of 5 will end in 5.  This means the units (or ones) digit will alternate

between 1 and 6 because we either have 6 or 11 at the end.

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If you wanted to find the actual sum of this the formula is:

 

N-1

 ∑rⁿ = (1-r^N)/(1-r)

n=0

 

2013

  ∑5ⁿ = (1-5²⁰¹⁴)/(1-5)

 n=0

 

= (1-5²⁰¹⁴)/(-4)

which is a really huge number which will be expressed in scientific notation