Evaluate $5+4\cdot 5+4\cdot5^2+4\cdot5^3+4\cdot5^4+4\cdot5^5$

Question

Evaluate $$5+4\cdot 5+4\cdot5^2+4\cdot5^3+4\cdot5^4+4\cdot5^5.$$ The options are $5^6$, $5^7$, $5^8$, $5^9$, $5^{10}$.

I'm new to this site. I came across this question in an Olympiad foundation site. I have no idea how to solve it. Can I get the answer of this question. Thanks.

Answer 1

Hint. Note that the given sum can be written as $$5+(5-1)\cdot 5+(5-1)\cdot5^2+(5-1)\cdot5^3+(5-1)\cdot5^4+(5-1)\cdot5^5.$$

Answer 2

$$a=5+4.5+4.5^2+4.5^3+4.5^4+4.5^5\to \times 5\\5a=25+4.5^2+4.5^3+4.5^4+4.5^5+4.5^6$$ now $5a-a= ?$ $$\quad{5a-a=(25-5)-(4.5)+(4.5^2-4.5^2)+(4.5^3-4.5^3)+(4.5^4-4.5^4)+(4.5^5-4.5^5)+4.5^6\\\to 5a-a=4.5^6 \\4a=4.5^6\\a=5^6}$$

Answer 3

$5 + 4.5 + 4.5^2+4.5^3+4.5^4+4.5^5 = 5+5(5-1)(1+5+5^2+5^3+5^4) = 5+5(5^5-1) = 5^6.$