Question

A) \(\left(25^6-15^6-10^6\right):5^6\)

B)\(1+2+2^2+2^3+...+2^{100}\)

C)\(7-7^4+7^5-...+7^{301}\)

Answer 1

a,

\(\left(25^6-15^6-10^6\right):5^6\\ =\left[\left(5\cdot5\right)^6-\left(3\cdot5\right)^6-\left(2\cdot5\right)^6\right]:5^6\\ =\left(5^6\cdot5^6-3^6\cdot5^6-2^6\cdot5^6\right):5^6\\ =5^6\left(5^6-3^6-2^6\right):5^6\\ =5^6-3^6-2^6\\ =15625-729-64\\ =14896-64\\ =14832\)

b,

\(1+2+2^2+...+2^{100}\\ =1\cdot\left(1+2+2^2+...+2^{100}\right)\\ =\left(2-1\right)\left(1+2+2^2+...+2^{100}\right)=\left(2-1\right)\cdot1+\left(2-1\right)\cdot2+\left(2-1\right)\cdot2^2+...+\left(2-1\right)\cdot2^{100}\\ =2-1+2^2-2+2^3-2^2+...+2^{101}-2^{100}\\ =2^{101}-1\)

Answer 2

B) 1+2+2²+2³+....+2

Tính B vs B=1+2+2²+2³+....+2⁹⁹+2¹⁰⁰

=>2B=2²+2³+2⁴+...+2¹⁰⁰+2¹⁰¹

=.2B-B=(2+2²+2³+2⁴+...+2⁹⁹+2¹⁰⁰

B=2¹⁰¹-1

=>B=2¹⁰¹-1