# Progression - 2

Illustration 5:

If x = 1+ a+ a^{2}+a^{3}+ --------+¥ and y = 1+b+b^{2}+ b^{3}+ ------ +¥ Show that,

_{ }

Where 0<a<1 and 0<b<1.

Solution:

Given x = 1+ a+ a^{2}+a^{3}+ --------+¥

_{ }

y = 1+b+b^{2}+ b^{3} ------ +¥

_{ }

Since 0<a<1, 0<b<1

\ 0<ab<1.

Now _{ }

_{ }

_{ }

**Arithmetic Geometric Series (A.G.P):
**Suppose a

_{1}, a

_{2}, a

_{3}, ------- a

_{n}be an A.P and b

_{1}, b

_{2}, b

_{3}, ------- b

_{n}is a G.P Then the sequence a

_{1}b

_{1}, a

_{2}b

_{2}, -------- a

_{n}b

_{n}is A.G.P.

A.G.P is of form _{ }

Where clearly _{ }

1) Sum of n terms of an A.G.P

_{ }

Why?

Let _{ }

Now if we multiply the series by r then.

_{ }

So on substraction,

_{ }

2) Sum of infinite series S_{¥}.

_{ }

Why?

If |r|<1 then _{ }

So, _{ }

Illustration 6:

If sum to infinity of the series _{ } then find x.

Solution:

We know that _{ }

Here a=1, b=1, r=x, d = 3

_{ }

Dumb Question:

1) Why_{ } ?

Ans: For infinite series to be summable |x| needs to be less than 1 hence _{ }

**Harmonic Progression (H .P):**

The sequence a_{1}, a_{2}, ------ a_{n} is said to be a H.P if _{ } is an A.P.

The n^{th} term of a H.P (t_{n}) is given by _{ } .

**Harmonic Means (H .M):**

If H_{1}, H_{2}, H_{3}-------- H_{n} be n H.M’s between a and b then a, H_{1}, H_{2}, H_{3}-------- H_{n}, b is a H.P.

This means _{ } is a A.P.

And hence _{ }

Note:

1) If a_{1}, a_{2}, a_{3}---------a_{n} are n non-zero numbers then H.M(H) of these number is given by

_{ }

2) If a, b, c are in H.P then _{ }

Why?

a, b, c are in H.P so, _{ } are in A.P

And hence _{ }

Illustration 7:

If the (m+1)^{th}, (n+1)^{th} and (r+1)^{th} term of an A.P are in G.P m, n, r are in H.P, Show that ratio of the common difference to the first term in the A.P is (-2/n).

Solution:

Let ‘a’ be the first term and ‘d’ be common difference of the A.P. Let x, y, z be the (m+1)^{th}, (n+1)^{th} and (r+1)^{th} term of the A.P then x = a+md, y = a+nd, z = a+rd. Since x, y, z are in G.P.

\ y^{2} = xz i.e. (a+nd)^{2} = (a+rd) (a+md)

_{ }

Now m, n, r in H.P

_{ }

_{ }

_{ }

**Some Important Theorems: **

If A, G, H are respectively AM, GM, HM between two positive unequal quantities then.

1) A>G>H

Why?

First of all let us Prove A>G.

The two numbers be x, y.

_{ }

So to prove _{ }

_{ }

Hence A>G -------- (1)

Now Let us Prove G>H

Again _{ }

_{ }

Combining (1) and (2) we get A>G>H.

Why G^{2}=AH?

Let x, y be two numbers.

So, _{ }

Hence

_{ }

Illustration 8:

If a, b, c, d, be four distinct positive quantities in H.P then show that a+d>b+c.

Solution:

a, b, c, d are in H.P

Then A.M > H.M

For first three terms _{ }

Ãž a+c>2b ------ (1)

And for last three terms

_{ }

Ãž b+d > 2c ------ (2)

From (1) and (2)

a+c+b+d > 2b+2c

Ãž a+d > b+c.