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3 years ago

Use the method of mathematical induction to prove that for each integer n≥1, 5+9+13+⋯+(4n+1)=2n^2+3n

Mathematics

1 answer:

Anastaziya [24]3 years ago

5 0

Answer:

Step-by-step explanation: 834^2n>4n

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Without a calculator how would you solve this?

mina [271]

Answer:

the largest number that is less than (2+√3)^6 is 2701

Step-by-step explanation:

$\bigstar \ (2+\sqrt{3})^6= \\\\C{}^{0}_{6}2^{6}+C{}^{1}_{6}2^{5}\sqrt{3}^1+C{}^{2}_{6}2^{4}\sqrt{3}^2+C{}^{3}_{6}2^{3}\sqrt{3}^3+C{}^{4}_{6}2^{2}\sqrt{3}^4+C{}^{5}_{6}2^{1}\sqrt{3}^5+C{}^{6}_{6}\sqrt{3}^6$

$=64+192\sqrt{3}+720 +480\sqrt{3} +540+108\sqrt{3} +27\\=1351+780\sqrt{3}$

$\bigstar\ (2-\sqrt{3} )^6 =\\\\1351-780\sqrt{3}$

$\bigstar\ \bigstar\ (2+\sqrt{3} )^6+(2-\sqrt{3} )^6 =\\\\1351+780\sqrt{3}+1351-780\sqrt{3}$

$\Longrightarrow(2+\sqrt{3} )^6+(2-\sqrt{3} )^6 =2702$

$\Longrightarrow(2+\sqrt{3} )^6 =2702-(2-\sqrt{3} )^6$

$0 < \left( 2-\sqrt{3} \right) < 1 \Longrightarrow 0 < \left( 2-\sqrt{3} \right)^{6} < 1 \Longrightarrow-1 < \left( 2-\sqrt{3} \right)^{6} < 0$

$\Longrightarrow2702-1 < 2702-\left( 2-\sqrt{3} \right)^{6} < 2702+0$

$\Longrightarrow 2701 < \left( 2+\sqrt{3} \right)^{6} < 2702$

8 0

3 years ago

Nandeville and Henningsburg are 75 miles apart. On a map, the two cities are 5 inches apart. How many miles does 1 inch represen

soldier1979 [14.2K]

Answer:

5 miles

Step-by-step explanation:

because when u divide 10 and 2 you will get 5

6 0

3 years ago

17. The length of a rectangle is 12 in, and the perimeter is 56 in. Find the width of the rectangle.

Ivahew [28]

Answer:

Step-by-step explanation:

P = 2(L + W)

P = 56

L = 12

now sub...we are looking for w

56 = 2(12 + W)

56 = 24 + 2W

56 - 24 = 2W

32 = 2W

32/2 = W

16 = W <==== the width is 16 inches

check...

P = 2(L + W)

56 = 2(12 + 16)

56 = 2(28)

56 = 56 (correct)

4 0

3 years ago

Please help me with this

Soloha48 [4]

Answer: - ∞

Step-by-step explanation:

As x approaches 4 from the positive side, the denominator gets closer to zero and the numerator gets closer to -2

-2/0 = -∞

5 0

3 years ago

Solve the system of equations. y=x+6y=x2+8x+16

Semenov [28]

Answer:

(- 2 ; 4) or (-5 ; 1)

Step-by-step explanation:

y = x + 6

y = x² + 8x + 16

x + 6 = x² + 8x + 16

x² + 8x + 16 - x - 6 = 0

x² + 7x + 10 = 0

x² + 5x + 2x + 10 = 0

x(x + 5) + 2(x + 5) = 0

(x + 2)(x + 5) = 0

x + 2 = 0 => x₁ = - 2

x + 5 = 0 => x₂ = - 5

y₁ = - 2 + 6 => y₁ = 4

y₂ = - 5 + 6 => y₂ = 1

6 0

4 years ago