Arithmetico Geometric Series and Harmonic Series

D. Arithmetico-Geometric Series

A series each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series . e.g. 1 + 3x + 5x² + 7x³ + .....

Here 1, 3, 5, .... are in AP & 1, x, x², x³ ..... are in GP .

Sum of n terms of an Arithmetico-Geometric Series

Let S_n = a + (a + d) r + (a + 2 d) r² + ..... + [a + (n - 1)d]  r^n-1

then S_n =  ,

Sum To Infinity : If |r| < 1 & n → ∞ then 

 rⁿ = 0 . S_∞ = .

Ex.25 Find the sum to n terms of the series

Also find the sum if it exist if n → ∞.

Sol. S =  ....(1)

 =  ....(2)

 =

=  -  =

S_n = 2 = . 

If n → ∞ then S_∞ =  = 2

 Ex.26 If positive square root of, ..... ∞ is  , find the value of  ' a '.

Sol. .  =

now  =  

and  =  ( use A G P)

 = = . 2³ ⇒ 

a =

E. Harmonic Progression (HP)

A sequence is said to HP if the reciprocals of its terms are in AP .

If the sequence a₁, a₂, a₃, .... , a_n is an HP then 1/a₁, 1/a₂, .... , 1/a_n is an AP & converse. Here we do not have the formula for the sum of the n terms of an HP . For HP whose first term is a & second term is b, the n^th term is t_n = . If a, b, c  are in HP           

⇒ b = or 

  = .

 Ex.27 If a, b, c are in H.P. then prove that a³b³ + b³c³ + c³a³ = (9 ac - 6 b²) a²c² .

Sol. =  ⇒  - = 0 .

Use p + q + r = 0   ⇒  p³ + q³ + r³ = 3 pqr

Ex.28 If  prove that a, b, c are in HP unless b = a + c.

Sol. ⇒ = 0                        

⇒  = 0

Let a + c = λ + = 0  

⇒ = 0

acλ - bλ² + b²λ + acλ - 2abc = 0  

⇒  2ac(λ - b) - bλ (λ - b) = 0

⇒ (2ac - bλ) (λ - b) = 0

⇒a, b, c are in H.P. or a + c = b.

Ex.29 The value of x y z is 55 or 354/55 according as the series  a, x, y, z, b  is an A.P. or H.P. Find the values of  a & b given that they are positive integers.

Sol. 

Let   a, x, y, z, b  are in A.P. ⇒   b = a + 4 d  

⇒ d = b-a/4

Similarly  when  a, x, y, z, b  are in H.P.    

In the 1st case 

In the 2nd case 

dividing  a³ b³ = 7³ ⇒   a = 7  ;  b = 1   or  a = 1  ;  b = 7