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Câu 3:
\(A=3+3^2+...+3^{100}\)
\(3A=3^2+3^3+...+3^{101}\)
\(3A-A=3^2+3^3+...+3^{101}-\left(3+3^2+...+3^{100}\right)\)
\(2A=3^{101}-3\)
Mà: \(2A+3=3^N\)
\(\Rightarrow3^{101}-3+3=3^N\)
\(\Rightarrow3^{101}=3^N\)
\(\Rightarrow N=101\)
Vậy: ...
Câu 1:
\(A=4+2^2+...+2^{20}\)
Đặt \(B=2^2+2^3+...+2^{20}\)
=>\(2B=2^3+2^4+...+2^{21}\)
=>\(2B-B=2^3+2^4+...+2^{21}-2^2-2^3-...-2^{20}\)
=>\(B=2^{21}-4\)
=>\(A=B+4=2^{21}-4+4=2^{21}\) là lũy thừa của 2
Câu 6:
Đặt A=1+2+3+...+n
Số số hạng là \(\dfrac{n-1}{1}+1=n-1+1=n\left(số\right)\)
=>\(A=\dfrac{n\left(n+1\right)}{2}\)
=>\(A⋮n+1\)
Câu 5:
\(A=5+5^2+...+5^8\)
\(=\left(5+5^2\right)+\left(5^3+5^4\right)+\left(5^5+5^6\right)+\left(5^7+5^8\right)\)
\(=\left(5+5^2\right)+5^2\left(5+5^2\right)+5^4\left(5+5^2\right)+5^6\left(5+5^2\right)\)
\(=30\left(1+5^2+5^4+5^6\right)⋮30\)
Tính tổng:a)3+3/5+3/25+3/125+3/625
b)M=4/3.7+4/7.11+4/11.15+...+8/95.99
c)N=1/2+1/6+1/12+1/20+...+1/90
Ta có : \(M=\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+.....+\frac{4}{95.99}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+......+\frac{1}{95}-\frac{1}{99}\)
\(=\frac{1}{3}-\frac{1}{99}\)
\(=\frac{32}{99}\)
a) 3A=1.2.3 + 2.3.3 + 3.4.3 +... + n.(n+1).3
=1.2.(3-0) + 2.3.(4-1) + ... + n.(n+1).[(n+2)-(n-1)]
=[1.2.3+ 2.3.4 + ...+ (n-1).n.(n+1)+ n.(n+1)(n+2)] - [0.1.2+ 1.2.3 +...+(n-1).n.(n+1)]
=n.(n+1).(n+2)
=>S=[n.(n+1).(n+2)] /3
b)
Nhân 4 vào hai vế ta được:
4A = 4.[1.2.3 + 2.3.4 + 3.4.5 + … + (n – 1).n.(n + 1)]
4A = 1.2.3.4 + 2.3.4.4 + 3.4.5.4 + … + (n – 1).n.(n + 1).4
4A = 1.2.3.4 + 2.3.4.(5 – 1) + 3.4.5.(6 – 2) + … + (n – 1).n.(n + 1).[(n + 2) – (n – 2)]
4A = 1.2.3.4 + 2.3.4.5 – 1.2.3.4 + 3.4.5.6 – 2.3.4.5 + … + (n – 1).n(n + 1).(n + 2) – (n – 2).(n – 1).n.(n + 1)
4A = (n – 1).n(n + 1).(n + 2)
A = (n – 1).n(n + 1).(n + 2) : 4.
3A=1.2.3 + 2.3.3 + 3.4.3 +... + n.(n+1).3
=1.2.(3-0) + 2.3.(4-1) + ... + n.(n+1).[(n+2)-(n-1)]
=[1.2.3+ 2.3.4 + ...+ (n-1).n.(n+1)+ n.(n+1)(n+2)] - [0.1.2+ 1.2.3 +...+(n-1).n.(n+1)]
=n.(n+1).(n+2)
=>S=[n.(n+1).(n+2)] /3
Ta có ; K = \(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+.....+\frac{1}{45}\)
\(=1+\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+....+\frac{2}{90}\)
\(=1+\left(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+.....+\frac{2}{9.10}\right)\)
\(=1+2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+.....+\frac{1}{9.10}\right)\)
\(=1+2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{9}-\frac{1}{10}\right)\)
\(=1+2\left(\frac{1}{2}-\frac{1}{10}\right)\)
\(=1+1-\frac{1}{5}\)(nhân phá ngoặc)
\(=2-\frac{1}{5}\)< 2
Vậy K = \(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+.....+\frac{1}{45}\)< 2
Bài 1 :
A = 12 + 22 + 32 +....+n2
A = 12 + 2.(1+1) + 3.(2 +1) + 4.( 3 +1) +.....+n(n-1 + 1)
A = 1 + 1.2 + 2 + 2.3 + 3 + 3.4 + 4 +.....+ n.(n-1) + n
A = ( 1 + 2 + 3 + 4 +....+n) + ( 1.2 + 2.3 + 3.4 +....+(n-1).n
A = (n+1).{(n-1):n+1)/2 +1/3.[1.2.3 +2.3.3 +.....+(n-1)n.3]
A = (n+1).n/2+1/3.[1.2.3 +2.3.(4-1)+ ...+(n-1).n [(n+1) - (n -2)]
A = (n+1)n/2+1/3.( 1.2.3 + 2.3.4 -1.2.3 +..+ (n-1)n(n+1)- (n-2)(n-1)n)
A =(n+1)n/2 + 1/3.(n-1)n(n+1)
A = n(n+1)[1/2 + 1/3 .(n-1)]
A = n.(n+1) \(\dfrac{3+2n-2}{6}\)
A= n.(n+1)(2n+1)/6
Bài 2 :
a, (x+1) +(x+2) + (x+3)+...+(x+10) = 5070
(x+10 +x+1).{( x+10 - x -1): 1 +1):2 = 5070
(2x + 11)10 : 2 = 5070
( 2x + 11)5 = 5070
2x+ 11 = 5070:5
2x = 1014 - 11
2x = 1003
x = 1003 :2
x = 501,5
b, 1 + 2 + 3 +...+x = 820
( x + 1)[ (x-1):1 +1] : 2 = 820
(x +1).x = 820 x 2
(x +1).x = 1640
(x +1) .x = 40 x 41
x = 40
\(A=1^2+2^2+3^2+....+10^2\\ A=1^{ }+\left(1+1\right)\cdot2+3\cdot\left(2+1\right)+.....+10\cdot\left(9+1\right)\\ A=1+2\cdot1+2+3\cdot2+3+....+10\cdot9+10\\ A=\left(1+2+3...+10\right)+\left(1\cdot2+3\cdot2+.....+10\cdot9\right)\)
Gọi 1+2+3+...+10 là P
Số số hạng là: (10 - 1) : 1 +1 = 10 (số)
P = (10+1) . 10 : 2 = 55
P = 55
Gọi \(1\cdot2+2\cdot3+....+9\cdot10\) là C
\(C=1\cdot2+2\cdot3+....+9\cdot10\\ 3\cdot C=1\cdot2\cdot3+2\cdot3\cdot3+....+9\cdot10\cdot3\\ 3\cdot C=1\cdot2\cdot3+2\cdot3\cdot\left(4-1\right)+....+9\cdot10\cdot\left(11-8\right)\\ 3\cdot C=1\cdot2\cdot3+2\cdot3\cdot4-1\cdot2\cdot3+.....+9\cdot10\cdot11-8\cdot9\cdot10\\ 3\cdot C=9\cdot10\cdot11\\ 3\cdot C=990\\ C=330\)
\(=>A=P+C\\ =>A=55+330\\ A=385\)
b)
\(B=5^2+10^2+15^2+...+50^2\\ B=5^2+\left(2\cdot5\right)^2+\left(3\cdot5\right)^2+....+\left(5\cdot10\right)^2\\ B=5^2+2^2\cdot5^2+3^2\cdot5^2+...+5^2\cdot10^2\\ B=5^2\cdot\left(1+2^2+3^2+....+10^2\right)\\ B=25\cdot\left(1+2^2+3^2+....+10^2\right)\)
\(\left(1+2^2+3^2+....+10^2\right)=A\)
\(=>B=25\cdot A\\ B=25\cdot385\\ B=9625\)
a) \(A=1^2+2^2+3^2+...+n^2\)
\(=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+n\left[\left(n+1\right)-1\right]\)
\(=\left[1\cdot2+2\cdot3+3\cdot4+...+n\left(n+1\right)\right]-\left(1+2+3+...+n\right)\)
\(=n\left(n+1\right)\left(n+2\right)-\frac{n\left(n+1\right)}{2}\)
\(=\left[n\left(n+1\right)\right]\left[\left(n+2\right)-\frac{1}{2}\right]\)
\(=n\left(n+1\right)\left(n+1,5\right)\)
còn câu b thì sao