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

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

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