Question

cho S=5+5²+5³+..........+5²⁰⁰⁶

a,Tinh S

b,Chung minh S chia het cho 126

Answer 1

S =  5 + 5² + 5³ + ......... + 5²⁰⁰⁶

5S = 5² + 5³ + 5⁴ + .......... + 5²⁰⁰⁷

5S - S = ( 5² + 5³ + 5⁴ + .......... + 5²⁰⁰⁷) - (  5 + 5² + 5³ + ......... + 5²⁰⁰⁶ ) 

4S = 5²⁰⁰⁷ - 5

S = \(\frac{5^{2007}-5}{4}\)

Answer 2

a)\(S=5+5^2+5^3+.....+5^{2006}\Rightarrow5S=5^2+5^3+5^4+\)\(....+5^{2007}\)

\(\Rightarrow5S-S=\left(5^2+5^3+5^4+....+5^{2007}\right)-\left(5+5^2+5^3+.....+5^{2006}\right)\)

\(\Rightarrow4S=5^{2007}-5\Rightarrow S=\frac{5^{2007}-5}{4}\)

Answer 3

\(a.S=5+5^2+5^3+......+5^{2006}\)
\(S=\left(5+5^2+5^3+5^4+5^5+5^6\right)+.....+\left(5^{2001}+5^{2002}+.....+5^{2006}\right)\)

\(S=5.\left(1+5+5^2+5^3+5^4+5^5\right)+......+5^{2001}\left(1+5+5^2+5^3+5^4+5^5\right)\)

\(S=5.3906+........+5^{2001}.3906\)

\(S=3906\left(5+....+5^{2001}\right)\)

\(b.S=3906\left(5+....+5^{2001}\right)\)

\(S=126.3\left(5+....+5^{2001}\right)\)

\(\Rightarrow\text{S chia hết cho 126}\)