Question

If x is the arithmetic mean of a and b, and y is the arithmetic mean of b and c, prove that:
(i) x + y = a + c
(ii) x = (3ac)/4

Answer 1

Correct Question

If x, y, z are in A.P, and x is the arithmetic mean of a and b, and y is the arithmetic mean of b and c, prove that ….

 

The arithmetic mean is deeply related to the series. Using the A.M, we can find a middle term in between any two values.

Then we know x is in between a and b, so y is in between b and c.

 

The A.P is a, x, b, y, c, and is in order.

Let the common difference be $d$.

A.P series: $b - 2d$, $b-d$, $b$, $b + d$, $b + 2d$

 

(i) Prove that $x+y=a+c$

$\implies (b-d)+(b+d)=(b-2d)+(b+2d)$

$\implies 2b = 2b$

 

(ii) is false because we cannot deduce equal numbers in both hands.

 

Learn More

The arithmetic mean(A.M) is related to the arithmetic progression(A.P), so are the G.M and G.P, and the H.M and H.P.

• G.M finds the mean of the exponent.
• H.M can find the middle term of two harmonic terms.

This is the G.M of two numbers, which is $\sqrt{ab}$.

This is the H.M of two numbers, which is $\dfrac{2ab}{a + b}$.

 

Questions

1. If two terms in G.P are given as $a_6 = 4$ and $a_{100} = 52$, what will be $a_{53}$?

(Ans. $a_{53} = 4 \sqrt{13}$, note there are even numbers of terms in between.)

2. If two terms in H.P are given as $a_7=60$ and $a_{11}=10$, what will be $a_9$?

(Ans. $a_9=\dfrac{120}{7}$)