we have
(1+ \frac{1}{x})^{x+1} = (\frac{x+1}{x})^{x+1}
=(\frac{x}{x+1})^{-(x+1)}
=(1- \frac{1}{x+1})^{-(x+1)}
= (1+ \frac{1}{-(x+1)})^{-(x+1)}
comparing with the RHS we get -(x+1) = n or x = -(n+1)
some short and selected math problems of different levels in random order I try to keep the ans simple
we have
(1+ \frac{1}{x})^{x+1} = (\frac{x+1}{x})^{x+1}
=(\frac{x}{x+1})^{-(x+1)}
=(1- \frac{1}{x+1})^{-(x+1)}
= (1+ \frac{1}{-(x+1)})^{-(x+1)}
comparing with the RHS we get -(x+1) = n or x = -(n+1)
Because 1995 is divisible by 3 so all the powers ≥2 of 1995 shall be divisible by 9.
So sum of digits of the result must be divisible by 9.
We get sum of digits of given number = 160 + X.
Smallest multiple of 9 above 160 is 162 which gives X= 2 and next multiple of 9 gives X = 11.
so X = 2
From the given condition we have
(x-3) = \sqrt[3]3 + \sqrt[3]{3^2}
squaring both sides
(x-3)^3 = (\sqrt[3]3)^3 + ( \sqrt[3]{3^2})^2 + 3 \sqrt[3]{3} \sqrt[3]{3^2}( \sqrt[3]{3} + \sqrt[3]{3^2}) using (a+b)^3 = a ^3 + b^3 + 3ab(a+)
or (x-3)^3 = 3 + 9 + 9 ( x-3) using ( \sqrt[3]{3^2} + \sqrt[3]{3^2} = x- 3
or (x^3 - 9x^2 + 27 x - 27 = 12 + 9x - 27 = 9x + 12
or x^3 - 9x^2+ 18x -12 = 0
We have numerator = 8^x - 2^x = (2^3)^x - 2^x = (2^(3x) - 2^x = 2^x(2^{2x} - 1)= 2^x(2^x + 1)(2^x-1)
Denominator = 6^x - 2^x = 3^x(2^x-1)
So \frac{8^x - 2^x }{6^x - 3^x} = \frac{2^x(2^x+1)}{3^x} = 2
Or 2^x(2^x+1) = 2 * 3^x
x cannot be zero as in the original question x = 0 is not possible
So we must ahve 2^x = 2 and 2^x+ 1 = 3^x by equating even on both sides and odd on both sides
1st equuation gives x = 1 and it satisfies 2nd equation as well.
so Solution x = 1
Let the sides be a -b , a , a+b
as the tringle is right angled so (a-b)^2 + a^2 = (a+b)^2
or a^2-2ab + b^2 + a ^2 = a^2 + 2ab + b^2
or a^2= 4ab
or a= 4b
as the triangls is right angled so aea = \frac{1}{2} a(a-b) = \frac{1}{2} 4b(4b-b) = 6b^2 = 24 or b= 2
the smallest sde = 3b = 6.
a,b,c are in HP so \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in AP
or \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}
multiply both LHS and RHS by a + b + c we get
\frac{a+b+c}{b} - \frac{a+b+c}{a} = \frac{a+b+c}{c} - \frac{a+b+c}{b}
Or
(\frac{a+b+c}{b} - 2) - (\frac{a+b+c}{a}-2) = (\frac{a+b+c}{c} -2) - (\frac{a+b+c}{b} -2)
or (\frac{a+c-b }{b}) - (\frac{b+c-a }{a}) = (\frac{a+b-c }{c}) - (\frac{c + a - b}{b} )
Hence \frac{b+c-a}{a}, \frac{c+a-b}{b}, \frac{a+b-c}{c} are in A.P
Let us put p = \sin ^2 \theta \cos^2 \theta
We have t_2 = \sin ^2 \theta + \cos^2 \theta = 1 \cdots(1)
and t_4 = (\sin^4x + \cos^4 x) = (\sin ^2 x + \cos^2 x) - 2 \sin ^2 x +\cos^2 x = 1 - 2p
Now we an sert
t_n t_2 = ( \sin ^n \theta + \cos^n \theta)(\sin ^2 \theta + \cos^2 \theta) = \sin ^{(n+2)}\theta + \cos^{(n+2)} + \sin ^n \theta \cos^2 \theta + \sin ^2 \theta \cos^n \theta
or t_n = t_{(n+2)} + t_{(n-2)}p
or t_(n+2) = t_n - t(n-2)p
Using this we have
t_6 = t_4 - t_2p = t_4 - p = 1- 3p
t_8 = t_6 - t_4p = 1-3p - (1-2p ) p^2 = 1- 4p + 2p^2
t_{10} = t_8 - t_6p = (1-4p +2p^2) - (1-3p)p = 1 - 5p + 5p^2
so 6t_{10} − 15t_8 + 10t_6 = 6(1-5p + 5p^2 ) - 15(1- 4p + 2p^2 ) + 10(1-3p) = 1
let the prime be p
we have 2019^8 \equiv -1 \pmod p
or 2019^{16} \equiv 1 \pmod p
so 2019^{16k} \equiv 1 \pmod p
as per Formats Little theorem p shall be of the form 16k + 1.
any number less than 16 is not possible because 2019^8 \equiv -1 \pmod p and 16=2^4 and 8=2^3 so any factor of 16 shall not give a power =1.
so p = 16 *2 + 1 = 33 (not a prime) or 16 * 3 + 1= 49( not a prime) or 16 * 4 + 1 = 65( not a prime) or 16 * 5 + 1 = 81 (not a prime) or 16 *6 + 1 = 97.
so we check for 97
now 2019 \equiv - 18 \pmod {97}
so 2019^2 \equiv 324 \pmod {97}
or 2019^2 \equiv 33 \pmod {97}
so 2019^4 \equiv 33^2 \pmod {97}
or 2019^4 \equiv 22 \pmod {97}
or 2019^8 \equiv 484 \pmod {97}
or 2019^8 \equiv -1 \pmod {97}
or or 2019^8 + 1\equiv 0 \pmod {97}
so smallest prime factor = 97
let f(x) = x^4+ 12x--5
As there change of sign once in f(x) so there is one positve real toot as per descartes' rule of signs
f(- x) = x^4- 12x--5
As there change of sign once in f(-x) so there is one real toot as per descartes' rule of signs
so there are 2 real roots
Number of ways of choosing r objects from p objects is \frac{(p!}{r!(p-r)!}. so \frac{(p!}{r!(p-r)!} is an integer
or \frac{p(p-1)!}{r!(p-r)!} is integer . Now because p does not divide the denominator because p is prime dividing by p we get the result