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Ta có: 3A = 3.(1+3+32+33+...+399+3100)
3A = 3+32+33+...+3100+3101
Suy ra: 3A – A = (3+32+33+...+3100+3101)−(1+3+32+33+...+399+3100)
2A = 3101−1
⇒ A = 3101−1
2
Vậy A = 3101−1
2
A=3+32+33+...+3100
3A=32+33+...+3101
3A-A=(32+33+...+3101)-(3+32+33+...+3100)
2A=3101-3
2A+3=3101
\(A=3+3^2+3^3+...+3^{100}\)
\(\Rightarrow3A=3.\left(3+3^2+3^3+...+3^{100}\right)\)
\(\Rightarrow3A=3^2+3^3+3^4+...+3^{101}\)
\(\Rightarrow3A-A=2A=\left[3^2+3^3+3^4+...+3^{101}\right]-\left[3+3^2+3^3+...+3^{100}\right]\)\(\Rightarrow2A=3^{101}-3\)
Theo đề bài ta có 2A + 3 = 3n ( \(n\in N\) )
\(\Rightarrow2A+3=3^{101}-3+3=3^n\)
\(\Rightarrow2A+3=3^{101}=3^n\)
\(\Rightarrow3^{101}=3^n\)
\(\Rightarrow101=n\) ( thỏa mãn điều kiện \(n\in N\)
Vậy n = 101
Ta có: A = 3 + 3 2 + 3 3 + . . . + 3 100
=> 3 A = 3 2 + 3 3 + 3 4 + . . . + 3 101
=> 3 A - A = ( 3 2 + 3 3 + 3 4 + . . . + 3 101 ) - ( 3 + 3 2 + 3 3 + . . . + 3 100 )
=> 2 A = 3 2 + 3 3 + 3 4 + . . . + 3 101 - 3 - 3 2 - 3 3 - . . . - 3 100
2 A = 3 101 - 3 <=> 2 A + 3 = 3 101 , mà 2 A + 3 = 3 n
=> n = 101
\(B=3^1+3^2+3^3+...+3^{100}\\3B=3^2+3^3+3^4+...+3^{101}\\3B-B=(3^2+3^3+3^4+...+3^{101})-(3^1+3^2+3^3+...+3^{100})\\2B=3^{101}-3\\\Rightarrow 2B+3=3^{101}\)
Mặt khác: \(2B+3=3^n\)
\(\Rightarrow 3^n=3^{101}\\\Rightarrow n=101(tm)\)
Vậy n = 101.
d)12.x - 33= 3 2 . 3 3
12.x - 33 = 9.27
12.x - 33 = 243
12.x = 243 + 33
12.x = 276
x = 276:12
x = 23
12x – 33 = 32.33
12x - 33 = 9.27
12x - 33 = 243
12x = 243 + 33
12x = 276
x = 276:12
x = 23
Vậy x = 23
\(B=1\cdot2+2\cdot3+3\cdot4+\dots+31\cdot32+32\cdot33\\3B=1\cdot2\cdot3+2\cdot3\cdot3+3\cdot4\cdot3+\dots+31\cdot32\cdot3+32\cdot33\cdot3\\=1\cdot2\cdot3+2\cdot3\cdot(4-1)+3\cdot4\cdot(5-2)+\dots+31\cdot32\cdot(33-30)+32\cdot33\cdot(34-31)\\=1\cdot2\cdot3+2\cdot3\cdot4-1\cdot2\cdot3+3\cdot4\cdot5-2\cdot3\cdot4+\dots+31\cdot32\cdot33-30\cdot31\cdot32+32\cdot33\cdot34-31\cdot32\cdot33\\=32\cdot33\cdot34\\\Rightarrow B=\dfrac{32\cdot33\cdot34}{3}=32\cdot11\cdot34\)
Vì \(32\cdot11\cdot34⋮34\)
nên \(B\vdots34\).
Ta có : B = 1.2 +2.3 +...+31.32 + 32.33
3B = 1.2.3 + 2.3.3+...+ 31.32.3 + 32.33.3
3B = 1.2.3 + 2.3.(4-1) +...+ 31.32.(33-30)+32.33.(34-31)
3B = 1.2.3 + 2.3.4 - 2.3.1 +...+ 31.32.33-30.31.32 + 32.33.34- 31.32.33
3B = 32.33.34 B = 32.33.34 / 3 = 32.11.34 chia hết cho 34
Vậy B chia hết cho 34
12x - 33 = 3².3³
12x - 33 = 9.27
12x - 33 = 243
12x = 243 + 33
12x = 276
x = 276 : 12
x = 23
\(12x-33=3^2\cdot3^3\)
\(12x-33=3^{2+3}\)
\(12x-33=3^5\)
\(12x-33=243\)
\(12x=243+33\)
\(12x=276\)
\(x=\dfrac{276}{12}\)
\(x=23\)