Question

Prove that 5 + 5² + 5³ + .... + 5^n = 5/4(5^n-1)
By mathematical induction​

Answer 1

Step-by-step explanation:

How can I prove by modular arithmetic that 5^(5n+1)+4^(5n+2)+3^(5n) is divisible by 11?

Want to migrate to Germany?

Exploit the laws of exponents, which apply to modular arithmetic as well as to regular integer arithmetic.

So let’s attack the first term

55n+1≡(55)n×51mod11≡(284×11+1)n×5mod11≡(1)n×5mod11≡5mod11

The second term:

45n+3≡(45)n×42mod11≡(93×11+1)n×42mod11≡(1)n×5mod11≡5mod11

The third term: (details not shown).

35n≡1mod11

So

35n+44n+2+35n≡5+5+1≡0mod11

Note: k5 can only have 3 possible values modulo 11:

k012345678910k5mod110110111101010110

Answer 2

Step-by-step explanation:

How can I prove by modular arithmetic that 5^(5n+1)+4^(5n+2)+3^(5n) is divisible by 11?

Want to migrate to Germany?

Exploit the laws of exponents, which

apply to modular arithmetic as well as to regular integer arithmetic.

So let's attack the first term

55n+1=(55)nx51mod11=(284x11+1)nx5mod1

1=(1)nx5mod11=5mod11

The second term:

45n+3=(45)nx42mod11=(93×11+1)nx42mod

11=(1)nx5mod11=5mod11

The third term: (details not shown).