Answer :
Answer:
See below
Step-by-step explanation:
the n number of value of x
[tex] \displaystyle x_{1},x _{2} \dots x_{n}[/tex]
let it be
[tex] \displaystyle x_{1} = x _{2} = x_{3}{\dots }= x_{n} = a[/tex]
now, the A.M of x is
[tex] \rm \displaystyle \: A.M = \frac{ x_{1} + x_{2} + \dots \dots \: + x_{n} }{n} [/tex]
since every value equal to a
substitute:
[tex] \rm \displaystyle \: A.M = \frac{ a + a + \dots \dots \: + a}{n} [/tex]
[tex] \rm \displaystyle \: A.M = \frac{ na}{n} [/tex]
reduce fraction:
[tex] \rm \displaystyle \: A.M = a[/tex]
the G.M of x is
[tex] \rm\displaystyle \: G.M =( x_{1} \times x _{2} {\dots }\times x_{n} {)}^{ {1}^{}/ {n}^{} } [/tex]
since every value equal to a
substitute:
[tex] \rm\displaystyle \: G.M =( a \times a{\dots }\times a{)}^{ {1}^{}/ {n}^{} } [/tex]
recall law of exponent:
[tex] \rm\displaystyle \: G.M =( {a}^{n} {)}^{ {1}^{}/ {n}^{} } [/tex]
recall law of exponent:
[tex] \rm\displaystyle \: G.M = a[/tex]
the H.M of x is
[tex] \displaystyle \: H.M = \frac{n}{ \frac{1}{ x_{1}} + \frac{1}{ x_{2} } {\dots } \: { \dots}\frac{1}{x _{n} } } [/tex]
since every value equal to a
substitute:
[tex] \displaystyle \: H.M = \frac{n}{ \frac{1}{ a} + \frac{1}{ a } {\dots } \: { \dots}\frac{1}{a } } [/tex]
[tex] \displaystyle \: H.M = \frac{n}{ \dfrac{n}{a} } [/tex]
simplify complex fraction:
[tex] \displaystyle \: H.M = n \times \frac{a}{n} [/tex]
[tex] \displaystyle \: H.M = a \: [/tex]
so
[tex] \displaystyle \: A.M = G.M = H.M = a[/tex]
hence,
[tex]\text{Proven}[/tex]