Question

Thu gọn tổng sau:
a)  A = 1+2+2²+2³+...+2⁶⁰
b)  B = 1+3+3²+3³+...+3⁴⁶
c)  C = 1+5²+5⁴+5⁶+...+5²⁰⁰
 

Answer 1

a) \(A=1+2+2^2+2^3+...+2^{60}\)

=>\(2A=2+2^2+2^3+2^4+...+2^{61}\)

=>\(2A-A=\left(2+2^2+2^3+2^4+...+2^{61}\right)-\left(1+2+2^2+2^3+...+2^{60}\right)\)

=>\(A=2^{61}-1\)

b)  \(B=1+3+3^2+3^3+...+3^{46}\)

=>\(3B=3+3^2+3^3+3^4+...+3^{47}\)

=>\(3B-B=\left(3+3^2+3^3+3^4+...+3^{47}\right)-\left(1+3+3^2+3^3+...+3^{46}\right)\)

=>\(2A=3^{47}-1\)

=>\(B=\frac{3^{47}-1}{2}\)

c) \(C=1+5^2+5^4+...+5^{200}\)

=>\(5^2C=5^2+5^4+5^6+...+5^{202}\)

=>\(25C=5^2+5^4+5^6+...+5^{202}\)

=>\(25C-C=\left(5^2+5^4+5^6+...+5^{202}\right)-\left(1+5^2+5^4+...+5^{200}\right)\)

=>\(24C=5^{202}-1\)

=>\(C=\frac{5^{202}-1}{24}\)

Answer 2

a) A = \(1+2+2^2+2^3+...+2^{60}\)

2A = \(2.\left(1+2+2^2+2^3+...+2^{60}\right)\)

2A = \(2+2^2+2^3+2^4+...+2^{61}\)

2A - A = \(\left(2+2^2+2^3+2^4+...+2^{61}\right)\)- \(\left(1+2+2^2+2^3+...+2^{60}\right)\)

A = \(2^{61}-1\)

b)B = \(1+3+3^2+3^3+...+3^{46}\)

3B = \(3.\left(1+3+3^2+3^3+...+3^{46}\right)\)

3B = \(3+3^2+3^3+3^4+...+3^{47}\)

3B - B = \(\left(3+3^2+3^3+3^4+...+3^{47}\right)\)- \(\left(1+3+3^2+3^3+...+3^{46}\right)\)

2B = \(3^{47}-1\)

B = \(\left(3^{47}-1\right):2\)

Answer 3

C nè bạn 

c ) C = \(1+5^2+5^4+5^6+...+5^{200}\)

5C = \(5^2\left(1+5^2+5^4+5^6+...+5^{200}\right)\)

5C = \(5^2+5^4+5^6+...+5^{202}\)

5C - C = \(\left(5^2+5^4+5^6+5^8+...+5^{202}\right)\)- (\(1+5^2+5^4+5^6+...+5^{200}\))

4C = \(5^{202}-1\)

C = \(\left(5^{202}-1\right):4\)