Question

Chứng minh:

            A = 3 + 3² + 3³ + ... + 3¹⁹⁹¹ 

a) chia hết cho 13

b) chia hết cho 41

MK cần gấp lắm giúp mk nha mk like cho!!!!!!!!!!!!!!!

Answer 1

Ta có: B= 3 + 3
3 + 3
5 + ... + 3
1991= ﴾3 + 3
3 + 3
5
﴿ + ﴾3
7+ 3
9 + 3
11
﴿ + ... + ﴾3
1987 + 3
1989 + 3
1991
﴿.
= 3 x ﴾1 + 3
2 + 3
4
﴿ + 3
7 x ﴾1 + 3
2 + 3
4
﴿ + ... + 3
1987 x ﴾1 + 3
2 + 3
4
﴿.
= 3 x 91 + 3
7 x 91 + ... + 3
1987 x 91= 3 x 7 x 13 + 3
7 x 7 x 13 + ... + 3
1987 x 7 x 13.
= 13 x ﴾ 3 x 7 + 3
7 x 7 + ... + 3
1987 x 7﴿.
Vì B = 13 x ﴾ 3 x 7 + 3
7 x 7 + ... + 3
1987 x 7﴿ nên B chia hết cho 13.
B= ﴾3 + 3
3 + 3
5 + 3
7
﴿ + ... + ﴾3
1985 + 3
1987 + 3
1989 + 3
1991
﴿.
= 3 x ﴾1 + 3
2 + 3
4 + 3
6
﴿ + ... + 3
1985 x ﴾1 + 3
2 + 3
4 + 3
6
﴿.
= 3 x 820 + ... + 3
1985 x 820= 3 x 20 x 41 + ... + 3
1985 x 20 x 41.
= 41 x ﴾ 3 x 20 + .. + 3
1985 x 20﴿
Vì B =41 x ﴾ 3 x 20 + .. + 3
1985 x 20﴿ nên B chia hết cho 41.

TK NHA

Answer 2

Ta có: B= 3 + 3 3 + 3 5 + ... + 3 1991= ﴾3 + 3 3 + 3 5 ﴿ + ﴾3 7+ 3 9 + 3 11 ﴿ + ... + ﴾3 1987 + 3 1989 + 3 1991 ﴿. = 3 x ﴾1 + 3 2 + 3 4 ﴿ + 3 7 x ﴾1 + 3 2 + 3 4 ﴿ + ... + 3 1987 x ﴾1 + 3 2 + 3 4 ﴿. = 3 x 91 + 3 7 x 91 + ... + 3 1987 x 91= 3 x 7 x 13 + 3 7 x 7 x 13 + ... + 3 1987 x 7 x 13. = 13 x ﴾ 3 x 7 + 3 7 x 7 + ... + 3 1987 x 7﴿. Vì B = 13 x ﴾ 3 x 7 + 3 7 x 7 + ... + 3 1987 x 7﴿ nên B chia hết cho 13.

B= ﴾3 + 3 3 + 3 5 + 3 7 ﴿ + ... + ﴾3 1985 + 3 1987 + 3 1989 + 3 1991 ﴿. = 3 x ﴾1 + 3 2 + 3 4 + 3 6 ﴿ + ... + 3 1985 x ﴾1 + 3 2 + 3 4 + 3 6 ﴿. = 3 x 820 + ... + 3 1985 x 820= 3 x 20 x 41 + ... + 3 1985 x 20 x 41. = 41 x ﴾ 3 x 20 + .. + 3 1985 x 20﴿ Vì B =41 x ﴾ 3 x 20 + .. + 3 1985 x 20﴿ nên B chia hết cho 41. 

Answer 3

A = 3 + 3² + 3³ + ... + 3¹⁹⁹¹

=> A=(3+3²+3³)+....+(3¹⁹⁸⁹+3¹⁹⁹⁰+3¹⁹⁹¹)

=> A=3.(1+3+3²)+....+3¹⁹⁸⁹.(1+3+3²)

=> A=3.13+....+3¹⁹⁸⁹.13

=> A=13.(3+...+3¹⁹⁸⁹)

=> A chia hết cho 13

còn câu b mk nghĩ là chia hết cho 40 thì ms đúng

Answer 4

a) Ta có: \(A=3+3^3+3^5+...+3^{1991}\)

\(=\left(3+3^3+3^5\right)+\left(3^7+3^9+3^{11}\right)+...+\left(3^{1987}+3^{1989}+3^{1991}\right)\)

\(=3\times\left(1+3^2+3^4\right)+3^7\times\left(1+3^2+3^4\right)+...+3^{1987}\times\left(1+3^2+3^4\right)\)

\(=3\times91+3^7\times91+...+3^{1987}\times91\)

\(=3\times7\times13+3^7\times7\times13+...+3^{1987}\times7\times13\)

\(=13\times\left(3\times7+3^7\times7+...+3^{1987}\times7\right)\)

Vì \(A=13\times\left(3\times7+3^7\times7+...+3^{1987}\times7\right)\)nên A chia hết cho 13.

b) Ta có: \(A=3+3^3+3^5+...+3^{1991}\)

\(=\left(3+3^3+3^5+3^7\right)+...+\left(3^{1985}+3^{1987}+3^{1989}+3^{1991}\right)\)

\(=3\times\left(1+3^2+3^4+3^6\right)+...+3^{1985}\times\left(1+3^2+3^4+3^6\right)\)

\(=3\times820+...+3^{1985}\times820\)

\(=3\times20\times41+...+3^{1985}\times20\times41\)

\(=41\times\left(3\times20+...+3^{1985}\times20\right)\)

Vì \(A=41\times\left(3\times20+...+3^{1985}\times20\right)\)nên A chia hết cho 41.