if a_k= 5 + 5^2 + 5^3 + 5^4 + ... +5^k

Question

If a_k= 5 + 5^2 + 5^3 + 5^4 + ... + 5^k , for what value of k will a_k be divisible by 10?

Answer 1

   a_k=5 + 5² + 5³ + 5⁴ + ... +5^k

5a_k=5² + 5³ + 5⁴ + ... +5^k+5^k+1

4a_k=5^k+1-5
a_k=(5^k+1-5)/4=5(5^k-1)/4

If we can show that (5^k-1)/4 is even for all k≥2 we are done. 

That shows that a_k is 5×an even number, i.e. divisible by 10.

(5²-1)/4=6 even, but (5³-1)/4=31.  

We try to show that for 2k, an even number and ≥2 that a_2k is divisible by 10.

We have result for 2k=2 above.

Assume that 5^2k-1 is a multiple of 8

Let 5^2k-1=8n and for k+1 we have

5^2(k+1)-1=5^2k+2-1=5²(5^2k-1)+5²-1=5²(8n)+24=multiple of 8.

 

 

Answer 2

5+5^2+5^3+5^4+...+5^k

5+25+125+625+...+5^k

(5+25)+(125+625)+...+5^k

30+750+5^5+5^6=30+750+3125+15,625=30+750+(3125+15,625)=

30+750+18750 is divisible by 10

looking at the pattern you want the next even power of 5 which is 5^6

starting with 5^1 you want consecutive pairs of powers of 5 to get numbers divisible by 10

(5^1+5^2)+(5^3+5^4)+(5^5+5^6)+(5^7+5^8)...