September 30, 2023

# Sequences and Series Tricks : Arithmetic and Geometric Sequences

Sequences and Series Tricks : Arithmetic and Geometric Sequences

Both arithmetic and geometric sequences  are  important concept of sequences and series . Questions from AP & GP are asked in various competitions including SSC and Railways.  Here, are some important formulas that you can use  to solve questions based on arithmetic and  geometric sequences quickly, easily and efficiently .

Formula 1.    If for an AP., sum of p terms is equal to sum of q terms then sum of (p+q) terms is zero.

Que. 1 If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is   the sum of the first 30 terms?
(1) 0                                               (2) –1
(3) 1                                               (4) Not unique

Sol. (i)   ( proper method)

Sum of 11 terms of an AP equals the sum of 19 terms of the same AP.

……….. (i)          and

…………….(ii

On equating  (1) and (2), we get

0r       . This implies

so          =  =  0.

(ii)( By using formula  )

Given that    .  So

Formula 2.   If the ratio of sum of ‘‘  terms of two arithmetic progression is     then the ratio of their ‘ ‘   term  will be  .

Que. 2 If the ratio of sum of  terms of two A.P. is  ,  then the ratio of      term is  :

(a)                 (b)                   (c)                       (d)

Sol.  (i) (proper method )

Let     and    be the first term and common difference  of first A.P.  and     and be the first term and common difference of second A.P. , then

Now put  . This implies .

So .

Hence

(ii)  By using formula:

Given

replace ‘n’ with 2n-1,we get

Now put n=12 we get

Formula 3: sum of infinite series of type  = where      and     are in A.P.  and

Que . 3     The sum of the following  series         is

(a)                                (b)  1                 (c)                            (d) none

sol.  (i) (proper method )  Let    be sum of the ‘n’ terms of   the given series , then

=

Taking limit     we get      = 3

(ii) (  by using formula)

Formula 4:  If  arithmetic mean and geometric mean of two numbers  ‘a’ and ‘b’  (a>b) are in ratio m:n , then

Que.4  Let     and     be positive real numbers such that   and , then   is equal to  (a)       (b)        (c)     (d)

Sol.  (proper  method)   Given that    which gives

By applying componendo and dividendo  =

By applying componendo and dividend again, we get

=    . So the option  (a) is correct .

(ii) (by using formula )  Given that      .

so    and .  Now by using formula 